Best hash algorithm to use in PHP/MYSQL

SHA-512 with a salt is a good & secure way to hash a password. If that's not available you have SHA-1 but it's security is considered a bit weak these days, especially if you don't use a salt.

There's a decent article here - short answer, use crypt(), and make sure you use a salt.

I would use the php's crypt() function because there will not be anyway for the password to be decrypted. When I need to check the newly entered password I just have to encrypt that one and compare the two results

There are a lot of options - see the php hash docs for the complete list.

Speed is not an advantage, so using sha-512 or whirlpool is a good idea. You don't have to store the full length of the hash in mysql, for instance you could hash something as whirlpool, which is 128 characters long, and store only the first 64 characters for efficiency.