In more detail
A hash function turns input of any size into output of a fixed size: 256 bits for SHA-256, 128 for MD5, 160 for SHA-1. Three properties matter. It is deterministic, so the same input always gives the same hash. It is one-way, so recovering the input from the hash is computationally infeasible. And it shows an avalanche effect: changing one bit of the input changes about half of the output bits. Hashes verify that data has not changed, identify duplicate files and are the building block of signatures and password storage. SHA-256 is the usual secure choice; MD5 and SHA-1 are broken for security use because collisions can be produced on purpose.
Why it matters
Published hashes let you confirm that a download is intact and untampered. Password databases should store salted, deliberately slow hashes rather than passwords. Treating a hash as encryption causes mistakes: there is no key and no way back, so a hash cannot "decrypt" a lost password or file. A fast hash such as plain SHA-256 is also a poor way to protect passwords on its own, because an attacker can test billions of guesses per second.
Example
The SHA-256 hash of hello is 2cf24dba5fb0a30e26e83b2ac5b9e29e1b161e5c1fa7425e73043362938b9824. Capitalise one letter, Hello, and the result is 185f8db32271fe25f561a6fc938b2e264306ec304eda518007d1764826381969: 125 of the 256 output bits differ. The length never changes; the MD5 hash of hello is always 32 hex characters, 5d41402abc4b2a76b9719d911017c592. Make your own with the Hash Generator, or hash a whole file with the File Checksum Calculator.
Common confusion
A hash is not Encryption: encryption is reversible with a key, hashing is not. A plain Checksum such as CRC32 catches accidental damage but is not built to resist deliberate tampering. For passwords, add a Salt so identical passwords do not produce identical hashes.