Ancient Egyptian is attached to the Hamito-Semitic language group (a name defined by linguists in the 18th century, more often called Afro-Asiatic by modern authors which gathers African and Middle East languages having common features in their vocabulary, their syntax and their grammatical forms, due either to a common origin, or to repeated contacts between populations.
Besides Egyptian, this family includes Cushitic languages (between the southern border of Egypt and the north of Tanzania) and perhaps Omotic (south of Ethiopia), semitic languages (Babylonian, Canaanite or Assyrian, later Aramean, Arab, Hebrew, etc), Chadic and Berber languages.
Many contradictory theories tried to define the origin of Egyptian language:
Came semitic people from the east to settle in the valley of the Nile where they dominated the native Hamitic populations?
Was Egyptian the common root of Hamitic and Semitic languages?
Has Egyptian a Sudanian or Ethiopian origin? Afro-centric theories even class Egyptian in the Negro-African language group which would be the origin of many modern African languages.
Today, many researchers believe that Afro-Asiatic could come from a language that was spoken in the area which is now Sahara, ca. 8000-6000 B.C. and spread in the north and the south when the former savanna turned into a desert...
In Afro-Asiatic languages have as commun feature that words inflect by changes around a root, usually made up of three consonants (sometimes two, sometimes four) which define the general concept. Suffixes, prefixes and additional vowels specify the type of the word (noun, verb...), its syntactic value and its meaning.
That’s why writing those languages does not require compulsorily vowels: only consonants and semi-consonants (wa, yu…) are written. Of course, this is today a problem for the modern pronunciation of hieroglyphs, since the use of the vowels has generally been forgotten, except when the words survived in Coptic or through other languages which had written them.
The evolution of Egyptian language
Of course, along its 4 500 years of use, the Egyptian language evolved in every aspect, sound, grammar and vocabulary. Obviously, even at the same time, the same language was not spoken in the same way from the north to the south of this large country, and it was influenced by the home area of the ruling dynasties which followed one another, with kings coming from one place or another.
No wonder thus that the usually described linguistics periods more or less follow Egypt’s great historical periods. The Egyptian languages known as of "the first phase" include first:
Archaic Egyptian, the language used during the predynastic and the thinite period;
Old Egyptian, used during the Old Kingdom and the first Intermediate Period. This dialect from the area of Memphis was the language of the texts of the Pyramids, inscriptions and documents of the 3rd to 6th dynasties of the Old Kingdom.
the Middle Egyptian was used during the Middle Kingdom and the second Intermediate Period. This language remained the Classical Egyptian. It was only spoken during 500 years (from about 2000 to 1500 B.C.) but remained afterwards the traditional language of hieroglyphic inscriptions during almost all the history of ancient Egypt. This language was used for many literary texts, royal and private inscriptions, administrative documents and letters as well as a numerous religious literature.
With the Egyptian languages known as “second phase" came new evolutions in every aspect (pronunciation, grammatical forms and use of the words, syntax, vocabulary, etc). they include:
The Late Egyptian (or neo-Egyptian), derived from dialects form Upper-Egypt, which replaced the middle Egyptian in the spoken language and was used during the New Kingdom and the third Intermediate Period, until ca. 600 B.C. It was written using hieroglyphic as well as hieratic script. It was the common language for non-literary texts from the 19th to the 25th dynasties, while the “classical” language remained in use during the same time. Amenhotep IV (Akhenaton) would have tried to adopt the Late Egyptian as the official language, since it often appears in the literary and even official documents during his reign.
At the Late Period, ca. 700 B.C., the languages spoken in the North and the South parted from one another, and so did the scripts. Under Psammetichus I, North and South were again unified. A new popular language appeared, the Demotic, which was used up to the 5th century A.D.
Coptic (the word comes from "aiguptos", the name the greek gave to Egyptians), appeared as of the 2nd century B.C., under the influence of the Greeks, the Romans, and later of the rise of the Christian religion. Spoken by the peasants of Upper-Egypt until the 17th century, it still remains the liturgical language of the Coptic church. But itself divided in several dialects (fayumic, akhmimic, bohairic in the north, and sahidic in the south).
Apparently, hieroglyphic writing appeared suddenly as a very advanced system as of 3000 B.C. But maybe, it was rather the result of a long development, performed on materials which did not survive up to now. We thus know it only from the period where the country got unified.
According to the ancient Egyptians, writing was of divine origin. The god Thot gave it to mankind. Anyway, it seems that the egyptian civilization developed its writing system without any external contribution. The drawings it uses indeed represent local animals, plants and practices. Besides, the attempts to link it with the cuneiform methods did not lead to convincing results.
Hieroglyphs
The first writing system is what we see on monuments and temples, royal tombs, religious and funerary texts, with an official and holy character: the hieroglyphic writing. Of course this writing evolved in order to comply with the requirements of the language’s evolutions, but it was nevertheless maintained throughout the Egyptian history, striving to maintain the tradition of the classical language.
The old Egyptians called their writing "medou netcher" (divine word). The signs were engraved in stone or painted on plaster, generally with much detail.
Hieroglyphs were used for writing old and middle Egyptian, but although middle Egyptian was no more talken about 1600 B.C., the writing of hieroglyphic inscriptions went on up to the end of the history of Ancient Egypt. During the Greco-Roman period, the number of signs increased, to reach the stage called Ptolemaic writing.
The last known hieroglyphic inscriptions were found on the island of Philae, and date from 394 A.D.
Everyday writing: Hieratic script
Too complicated and long to draw, the hieroglyphic writing was very early transformed for daily use, using signs which were faster and easier to handle.
When writing on papyrus or other material, it was impossible to keep the degree of detail which appeared on official inscriptions. A more cursive form thus developed very early.
It was later called "hieratic" (sacred), although it was initially intended for daily use. It used very simplified signs, which allowed to write and take notes quickly, but where the original graphic can hardly be recognized – somehow like our cursive script compared to printed block letters. Hieratic was written from right to left, in lines or columns.
From the New Kingdom, hieratic writing drifted, and an official from Lower Egypt had more and more difficulties reading a text written by a scribe in Upper Egypt in "abnormal hieratic". Along with time, scribes developped a second simplified form, the demotic script, and hieratic was then devoted only to religious texts – this explaining the name of Hieratic.
Demotic writing
Demotic appeared in Upper Egypt and was introduced as an official script during the 7th century B.C. The usual words were shortened, others were written with uniliteral signs used like an alphabet.
The first demotic texts appeared about 650 B.C. This script was used for everyday documents (administrative, legal, economic), then in literature and scientific texts, whereas hieratic was maintained for religious texts and hieroglyphs for monumental inscriptions.
In the Greco-Roman period, only priests could still read hieroglyphs. This is why the Rosetta Stone, for example, bears the same text in hieroglyphs and demotic, and also in greek, the official language of the Ptolemies. When the Roman emperor Theodosius, in 384, had the temples and thus the priest schools closed, the Pharaonic Egyptian scripts sank into oblivion. .
Coptic writing
When the Ptolemies seized power, Egyptian people were speaking and writing demotic. However, Greek became the official language. About 200 B.C., the greek alphabet got adapted to Egyptian: seven signs were added, taken from demotic, in order to represent sounds which did not exist in Greek. Thus, at the same time, the vowels were introduced in writing : the Coptic alphabet gathered 31 letters and used block and small letters.
At the beginning, Coptic script was used first for the transliteration in Greek of formulas and names, of magic or astronomical texts, in order to respect the way they had to be pronounced. At the time of the christianization of Egypt, in the 4th century, the religious literature was written in coptic script, since demotic remained still associated with the ancient beliefs. After the Muslim conquest in th 7th century, Coptic language and writing declined and finally disappeared in daily use, replaced by Arabic.
One must have approached training to hieroglyph reading to discover all its richness and the place it leaves to magics, pharaonic values, art and intuition; most of all, you have to leave aside all the logic you are used to.
It’s easy to imagine the efforts which were necessary for the first researchers, and the genius of Champollion, to decipher a writing method which included hundreds of signs (approximately 750 “usual” signs), aligned sometimes from the top to the bottom, from right to left or from left to right, without separation between words.
Moreover, the hieroglyphs have a magic significance: pronouncing them makes them alive, and thus some signs, whose reading could be dangerous, are sometimes voluntarily mutilated. Even their color has a magic meaning, and the scribe had to be careful not to use one which may offend a god.
Lastly, the sequence of the signs is not always that of reading: some royal or honorary signs are located in front of the word even if they are read at the end. And they are arranged so that the sentence is beautiful to look at, even if this means to mistreat the usual reading direction.
Some principles
the layout must be pleasant to look at. To reach this, the signs are often grouped by 3 or 4 in virtual "square blocks", called quadrats. The dimensions of the subject of the pictograms are not taken into account: an owl may be taller than a man.
the direction of reading is determined by looking at the signs which represent living beings: if the animals or men are looking to the left site, reading occurs from left to right, and vice-versa. This allow to decorate in a symmetrical way the front of buildings, by writing the same inscription in both directions.
" cartouches" (curvilinear frames) are intended to contain royal names.
Lastly, the hieroglyphs may have varied functions:
A hieroglyph can have the value of a pictogram (it was especially the case in the oldest writings), That is to say it means what it represents: (Per) a house, (Ra:) god-sun, (ra) the mouth. In this case, it is usually accompanied by a vertical bar.
It can be also an ideogram, that is to say it symbolizes a different object: a mouth for the speech, the sun as a star.
It can also be an image of the sound it represents, like in a rebus. These "sound signs” (or phonograms) include uniliteral signs, which evoke a single sound, like the consonants in our alphabet (= R), biliteral (two successive consonants, = PR), or triliteral signs (three successive consonants).
Sometimes "sound-signs" are there only to confirm the pronunciation of the sign which stands in front: in this case again, they are not pronounced...
Lastly, the same sign (sun) can be sometimes a pictogram (god-sun), sometimes an ideogram (sun), sometimes a sound-sign (Ra), sometimes a determinative (the word which precedes is relating to time).
In short: the training is enthralling, and accessible to those with really wish it, but not particularly simple!
If Egyptian writing looks quite complex, the representation of numbers seems to us surprisingly easy.
First of all, the numeral system is decimal.
There is no zero, but a sign exists for each "rank" : a bar for the unit, a bridge for tens, a loop for hundreds, etc.
In order to write a number, each sign (thousands, hundreds, tens and units) is drawn as many times as required. Reading thus results from a mental addition of the signs.
Thus the order in which the signs are written does not have any significance, and they can be arranged in a way which facilitates reading and is pleasant to look at...
However, the numbers have also a name, which can be written out using hieroglyphs!
Writing fractions deserves a special paragraph:
The Egyptians used for that a magic sign, the famous "eye of Horus" (called wadjet, wedjat or udjat) which is often used as an amulet. When Horus, the god Falcon, son of Isis and Osiris fought his uncle Seth to avenge his father, Seth snatched one of his eyes and cut it into pieces, but Thot, the healing god, reconstituted it and returned it to him. This eye consequently became a talisman which symbolizes health, light, and protection against... the "evil eye”!
Indeed, the various parts of the eye’s drawing are used to write the binary fractions (1/2, 1/4, 1/8, 1/16, 1/32 and 1/64). As for the figures, the other fractions are written by using several of these signs (addition of fractions). For example 3/8 = 1/4 + 1/8
Today, many researchers believe that Afro-Asiatic could come from a language that was spoken in the area which is now Sahara, ca. 8000-6000 B.C. and spread in the north and the south when the former savanna turned into a desert...
No wonder thus that the usually described linguistics periods more or less follow Egypt’s great historical periods. The Egyptian languages known as of "the first phase" include first:
The first writing system is what we see on monuments and temples, royal tombs, religious and funerary texts, with an official and holy character: the hieroglyphic writing. Of course this writing evolved in order to comply with the requirements of the language’s evolutions, but it was nevertheless maintained throughout the Egyptian history, striving to maintain the tradition of the classical language.
When writing on papyrus or other material, it was impossible to keep the degree of detail which appeared on official inscriptions. A more cursive form thus developed very early. 
Demotic appeared in Upper Egypt and was introduced as an official script during the 7th century B.C. The usual words were shortened, others were written with uniliteral signs used like an alphabet. 
When the Ptolemies seized power, Egyptian people were speaking and writing demotic. However, Greek became the official language. About 200 B.C., the greek alphabet got adapted to Egyptian: seven signs were added, taken from demotic, in order to represent sounds which did not exist in Greek. Thus, at the same time, the vowels were introduced in writing : the Coptic alphabet gathered 31 letters and used block and small letters. 
One must have approached training to hieroglyph reading to discover all its richness and the place it leaves to magics, pharaonic values, art and intuition; most of all, you have to leave aside all the logic you are used to. 
(Per) a house, 
(Ra:) god-sun, 
(ra) the mouth. In this case, it is usually accompanied by a vertical bar.
