The language of Old English, or Anglo-Saxon, developed from the language brought to the British Isles by the Germanic tribes that came there in the 5th century from the Continent: The Saxons, who had their own long history and whose language is essentially "Low German", the Angles, a tribe originating from the peninsula of Angeln in northeastern Germany, named after the predominant occupation of fishing (hence the modern English word "angling"), and, according to Bede, the Jutes (from Jutland, or Jylland, in Denmark). It has been speculated that there must have also been some Frisians in the migration, since a) they were in the way, living in the coastal areas of the North Sea, and b) even today, Frisian is the closest language to English.

Given this origin, Old English is therefore in the West Germanic group, along with Old Saxon, Old Low Franconian (the language of the Franks and ancestor of Dutch), and Old High German. Yet once on the Island, the language of the Angles and Saxons began to develop away from that of their continental cousins. Old English had influences from the North Germanic (Scandinavian) tribes that the contentinal Saxons did not have, while Saxon itself was influenced by the surrounding dialects of Franconian and High German.

The dialects of Old English can be identified roughly according to the political divisions on the Island. The three kingdoms that came to predominate were in the west, where there was more room for expansion/conquest: Northumbria, Mercia, and Wessex. Theirs were the main dialects, though there were other dialects, like Kentish and East Anglian. Thanks to King Alfred's political and military resilience against the Danes in the 9th century, and his cultural and religious efforts in copying works of literature and translating religious works from Latin into the vernacular, the dialect of West Saxon is the standard dialect for learning Old English.

The first instance of written Anglo-Saxon is Caedmon's hymn, preserved in Bede's Ecclesiastical History. There are many versions, but I believe the original was Northumbrian. Anyway, it's short, so here is the hymn in West Saxon, then Northumbrian, then Modern English.

West Saxon

Nu sculon herigean       heofonrices weard,
meotodes meahte       and his modgeþanc,
weorc wuldorfæder,       swa he wundra gehwæs,
ece drihten,       or onstealde.
He ærest sceop       eorðan bearnum
heofon to hrofe,       halig scyppend;
þa middangeard       moncynnes weard,
ece drihten,       æfter teode
firum foldan,       frea ælmihtig.

Northumbrian

Nu scylun hergan       hefaenricaes uard,
metudæs maecti       end his modgidanc,
uerc uuldurfadur,       sue he uundra gihuaes,
eci dryctin,       or astelidæ.
He aerist scop       aelda barnum
heben til hrofe,       haleg scepen;
tha middungeard       moncynnæs uard,
eci dryctin,       æfter tiadæ
firum foldu,       frea allmectig.

Modern English

Now we should praise the keeper of Heaven's kingdom,
the might of the Creator, and his thought,
the work of the Father of glory, how each of wonders
the Eternal Lord established in the beginning.
He first created for the sons of men
Heaven as a roof, the holy Creator,
then Middle-earth the keeper of mankind,
the Eternal Lord, afterwards made,
the earth for men, the Almighty Lord.