bcrypt is a hashing algorithm which is scalable with hardware (via a configurable number of rounds). Its slowness and multiple rounds ensures that an attacker must deploy massive funds and hardware to be able to crack your passwords. Add to that per-password salts (bcrypt REQUIRES salts) and you can be sure that an attack is virtually unfeasible without either ludicrous amount of funds or hardware.

bcrypt uses the Eksblowfish algorithm to hash passwords. While the encryption phase of Eksblowfish and Blowfish are exactly the same, the key schedule phase of Eksblowfish ensures that any subsequent state depends on both salt and key (user password), and no state can be precomputed without the knowledge of both. Because of this key difference, bcrypt is a one-way hashing algorithm. You cannot retrieve the plain text password without already knowing the salt, rounds and key (password). [Source]

How to use bcrypt:

Using PHP >= 5.5-DEV

Password hashing functions have now been built directly into PHP >= 5.5. You may now use password_hash() to create a bcrypt hash of any password:

<?php
// Usage 1:
echo password_hash('rasmuslerdorf', PASSWORD_DEFAULT)."\n";
// $2y$10$xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
// For example:
// $2y$10$.vGA1O9wmRjrwAVXD98HNOgsNpDczlqm3Jq7KnEd1rVAGv3Fykk1a

// Usage 2:
$options = [
  'cost' => 11
];
echo password_hash('rasmuslerdorf', PASSWORD_BCRYPT, $options)."\n";
// $2y$11$6DP.V0nO7YI3iSki4qog6OQI5eiO6Jnjsqg7vdnb.JgGIsxniOn4C

To verify a user provided password against an existing hash, you may use the password_verify() as such:

<?php
// See the password_hash() example to see where this came from.
$hash = '$2y$07$BCryptRequires22Chrcte/VlQH0piJtjXl.0t1XkA8pw9dMXTpOq';

if (password_verify('rasmuslerdorf', $hash)) {
    echo 'Password is valid!';
} else {
    echo 'Invalid password.';
}