Collision resistance is a property of cryptographic hash functions: a hash function is collision resistant if it is hard to find two inputs that hash to the same output; that is, two inputs a and b such that H(a) = H(b), and a ≠ b.[1]:136
Every hash function with more inputs than outputs will necessarily have collisions. Consider a hash function such as SHA-256 that produces 256 bits of output from an arbitrarily large input. Since it must generate one of 2256 outputs for each member of a much larger set of inputs, the pigeonhole principle guarantees that some inputs will hash to the same output. Collision resistance doesn't mean that no collisions exist; simply that they are hard to find.
The birthday "paradox" places an upper bound on collision resistance: if a hash function produces N bits of output, an attacker who computes "only" 2N/2 hash operations on random input is likely to find two matching outputs. If there is an easier method than this brute force attack, it is typically considered a flaw in the hash function.[1]:141
Cryptographic hash functions are usually designed to be collision resistant. But many hash functions that were once thought to be collision resistant were later broken. MD5 and SHA-1 in particular both have published techniques more efficient than brute force for finding collisions.[2][3] However, some compression functions have a proof that finding collisions is at least as difficult as some hard mathematical problem (such as integer factorization or discrete logarithm). Those functions are called provably secure.
Collision resistance is desirable for several reasons.