Who were the Anglo-Saxons, and what was Old English?

"The Anglo-Saxons" is the general name given to the Germanic peoples who inhabited Britain between the fifth and the eleventh centuries, between the Romans and the Normans. The name isn't a modern invention: it was first used in England at the court of Alfred the Great (871-899), who came to the throne as King of the West Saxons, but redefined his title as King of the Anglo-Saxons (rex Angolsaxonum) in the 890s, to mark his rulership over all free English people. It was used abroad even earlier, in the time of Charlemagne (768-814), but there it seems to have been to distinguish the "English" Saxons from those who stayed behind on the Continent. (Later on, the Anglo-Saxon Chronicle sometimes makes the same distinction, using "Old Saxons" to refer to the people of Germany.)

"Old English" is the name modern scholars give to the language of the Anglo-Saxons, though some scholars use "Anglo-Saxon" to refer to the language as well as the people. The Saxons themselves called their language Englisc (Old English -sc is pronounced like modern -sh, so they would have pronounced it "English"), and a lot of the low-level structure and vocabulary of our modern English goes back to their Englisc. The main effect of the Norman conquest in the long run was to add an extra layer of vocabulary.