The most common hash function is MD5. The MD5 hash of "hash" is "258def5e78a5f18e3477fcfc55104f2e". That will always be the hash of "hash".
Changing a minor detail of the hashed value will completely change the hash. The hash of "hash1" is "00c6ee2e21a7548de6260cf72c4f4b5b".
You can also hash a program or file on your computer. Everything on your computer can be broken down into binary(1s and 0s) and binary can always be converted into hexadecimal. The hexadecimal value is what is hashed, I believe.
You can see that this is very useful for verifying the integrity of files. If even 1 byte of data is out of place, the hash will be completely different. You will know that you must download the file again.
All P2P programs like limewire use hashes. Before it downloads the file, it downloads the hash. After the file is done, it checks it against the hash. If it does not match, it downloads both again, and repeats this until they do match.
Many large or secure downloads supply you with a hash. If you check against the hash, you can be 100% sure that part of it wasn't lost along the way, and that it is the original unmodified version.
Hashes are also used in security. A hash function only goes one way. There is no way to tell that "258def5e78a5f18e3477fcfc55104f2e" is the hash of "hash" without trying to hash every possible combination of letters until you get to that. This is very useful for storing passwords. In windows and (hopefully) advicenators, your passwords are stored as hashes. Someone can very easily get to your hashes, but if your passwords of sufficient length, it could take weeks for them to figure out your password.
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