Mementos are most easily described as special collectables. There are a variety of Mementos attainable, and a variety of types.
Mementos can be located in Catacombs > Passage 1, Private Chambers (Darkside Mementos).
Some Mementos offer unique, permanent buffs to your character's attack and defense stats. This type of Memento is typically attained by participating in Live Gaming Events.
Other mementos are momentous marks of your ingame actions. These are usually only for display, and attained by Fusion.