Cryptographic Hash Functions

These cryptographic checksums are called hash functions.
Common examples: MD5, SHA1

Hash functions generally have two properties:
One-way:
Given a hash value, it's hard to find a message which hashes to that value (a "preimage").
Collision-resistant:
It's hard to find two messages which hash to the same value (a "collision").

What is "hard"?
For SHA1, for example, it takes 2^160 work to find a preimage, and 2^80 work to find a collision.
The difference is due to the well-known birthday paradox.