The Animus Data Fragments are little pieces of data that, when collected, allow Desmond to explore areas of the Animus through a puzzle-like minigame.
In order to unlock each of the 5 Desmond Memories, you’ll need to collect increasing numbers of Fragments. 5, 10, 15, 20 and 30 Fragments each unlocks a Desmond Memory, and completing Desmond memories will mark the locations of a few more Fragments on your map, making them fairly easy to obtain.
Also, when you reach the 50 Fragment mark, the remaining 50 will appear on your map (as long as you have vision via viewpoints and capturing Dens).
The exception to this is of course the Fragments in Cappadocia, which need to be found separately.
When you get them all you’ll recieve the "Capped" Trophy/Achievement.
The Desmond Memories
These are mostly just segments in which Desmond’s monologue provides insight into his childhood and life before the events of Assassin’s Creed. They’re physics-based puzzling sessions (in the broadest possible sense) in a first person perspective, and generally consist of some simple platforming and the use of shapes to build bridges to the objective.
They’re very easy to get through, and text descriptions cannot do them justice. Treat them like an interactive cutscene, if you will. Keep your eyes open for randomly appearing multiplayer bonuses too, if you use uPlay, such as skins and titles for your character.
The Trophies/Achievements unlocked by completing them are listed below.