From 1d4chan

Aberration is a classification of monster introduced to the Dungeons & Dragons lexicon during third edition. Designed to help assist in collating the vast array of monsters into more useful groups, the Aberration classification ultimately came to be the dustbin of monster categories. All the freakish and weird and extra-gribbly monsters ultimately ended up being classified as aberrations, a tradition that continues to this day. The book Lords of Madness was a book dedicated to aberrations which went into great detail about the lore of them.

The distinction between Beast, Aberration, and Monstrosity is hard to define sometimes. The 5th Edition Monster Manual got into it a little bit, but a general rule is that Beasts came into being without magic, Aberrations can only exist because of magic and psionics, and Monstrosities are sort of the catch-all of the rest. With very few exceptions, Aberrations are (or used to be) intelligent, where Beasts and Monstrosities are not.

Pro-tip: if it looks like it could kind of come out of the works of H.P. Lovecraft, or is a direct rip-off of those critters, it's probably an aberration. Psionic powers are also a pretty common theme, too. Many but not all Abberations originated from the Far Realm.

Aberrations from D&D in include:

Pathfinder also has a few of its own aberrations: