Definition
Identity is the relation each thing bears to itself and to nothing else. It is what 'is the same as' expresses when we say the morning star is the same object as the evening star, or that Mark Twain is Samuel Clemens. To assert an identity is to claim that two names or descriptions pick out a single thing.
Identity sounds trivial — everything is itself — but it has rich logical structure. It is reflexive, symmetric, and transitive, and it obeys a powerful further principle: whatever is true of a thing is true of it under any name.
Why it matters
How it works
The governing principle is Leibniz's Law, the indiscernibility of identicals: if a and b are the very same object, then anything true of a is true of b. This is what makes 'the morning star is the evening star' so useful — once established, every fact about the one transfers to the other.
The puzzles begin with identity over time. A thing persists through change: you are the same person you were a decade ago, though scarcely a cell remains. But Leibniz's Law seems to say identical things share all properties — so how can one thing have incompatible properties at different times? Tense logic eases this by relativising properties to times. The harder cases, where the very parts are swapped out one by one, lead straight to the ship of Theseus.