Difference between collision resistance and target collision resistance For a hash function, what's the difference between Collision Resistance and Target Collision Resistance?. 

I understand the definition of hash function collision resistance, but I don't know about Target Collision Resistance.
Pre-image resistant but not 2nd pre-image resistant? describes the relationship between the three basic hash function security notions: Collision Resistance, Second Preimage Resistance and Preimage Resistance.

In short, Collision Resistance implies Second Preimage Resistance (but not vice-versa) - there is a good diagram on page 4 of RogawayShrimpton04 that illustrates the relationships.

The notion of Target Collision Resistance (which was introduced in BellareRogaway97) is a type of Second Preimage Resistance, and is implied by Collision Resistance.

More formally, Target Collision Resistance describes the ability of an adversary to find a second preimage for a fixed message (compared to the more general notion of Second Preimage Resistance, where the adversary to find a second preimage for a randomly chosen message).

These notions actually sound really similar (and probably in practice are) if you think about real world hash functions, but make more sense when you consider the security notions (and proofs of them) use keyed hash functions - in TCR, the adversary tries to find second preimages for a fixed message over multiple experiments with random keys, while in general 2PIR the message and the key are random in every experiment.