Where did the name "English" come from?

Supposedly, it is a corruption of Anglish, aka: the language of the Angles. They were a Germanic tribe that invaded/colonized Britain after the Romans legions left, in concert with the Saxons, Jutes, and Frisians.

Oddly (and perhaps confusingly), the name the Celtic-speaking residents used to refer to the invaders (and later Normans used for their English-speaking subjects) was "Saxons". Often formally "Anglo-Saxon" was used instead. Also, the modern language found to be closest to English is Frisian. Sadly, the Jutes get no such lasting recognition, except from historians.

This map from Wikipedia shows the supposed original invasion area in Kent, with a proposed tribal makeup.

The speculation is that the language started out as a sort of pidgin between the various German tribal languages. This is the theory for why the the word gender rules that a lot of other Germanic languages require were dropped.

According to Dictionary.com:

Origin: before 900; Middle English; Old English Englisc, equivalent to Engle (plural) the English (compare Latin Anglī; see Angle) + -isc -ish

And for "Angle" (link above):

noun: a member of a West Germanic people that migrated from Sleswick to Britain in the 5th century a.d. and founded the kingdoms of East Anglia, Mercia, and Northumbria. As early as the 6th century their name was extended to all the Germanic inhabitants of Britain.

Origin: < Old English Angle plural (variant of Engle ) tribal name of disputed orig.; perhaps akin to angle 2 if meaning was fisher folk, coastal dwellers

Although @drm65 and @T.E.D. have given correct answers to the original question, I thought I would deal with the “not Angles but Angels” reference raised in the comments on the question by @Joe Blow.

The “not Angles but Angels” story has no bearing on the derivation of “English,” but it is a real story related by Bede in The Ecclesiastical History of the English People, book 2, chapter 1, near the end of the chapter. Bede relates a story concerning Pope Gregory the Great encountering some English slaves for sale in the marketplace at Rome (this was before he became Pope himself, though). When told that they were pagans, he said:

‘Heu, pro dolor!’ inquit, ‘quod tam lucidi uultus homines tenebrarum auctor possidet, tantaque gratia frontispicii mentem ab interna gratia uacuam gestat!’ Rursus ergo interrogauit, quod esset uocabulum gentis illius. Responsum est, quod Angli uocarentur. At ille: ‘Bene,’ inquit; ‘nam et angelicam habent faciem, et tales angelorum in caelis decet esse coheredes. (Ecclesiastical History II.i)

Which translates as:

“Alas! what pity,” said he, “that the author of darkness is possessed of men of such fair countenances; and that being remarkable for such graceful aspects, their minds should be void of inward grace.” He therefore again asked, what was the name of that nation? and was answered, that they were called Angles. “Right,” said he, “for they have an Angelic face, and it becomes such to be co-heirs with the Angels in heaven.” (Source, translator uncertain but likely L.C. Jane’s 1903 edition)

The exact phrase “non angli sed angeli” does not appear in the original Latin as you can see; it is a later distillation of the basic sentiment into a more succinct construction.

The original is much more fun than this translation; it’s basically an opportunity for Gregory — or for Bede — to show off his Latinity by making some amusing puns. I’m not fully familiar with how the story has been received in the centuries since Bede wrote, but as Joe Blow’s reference demonstrates, it retains some limited currency even today.

Lastly, note that it is not certain that the incident occurred, or if it did, that this is an accurate report. Fair and unbiased reporting of history was not a high priority for Bede, who was after all writing an ecclesiastical history connecting England more firmly to the rest of Christendom.