<p>What is the fastest, yet secure way to encrypt passwords (in PHP preferably), and for whichever method you choose, is it portable?</p>

<p>In other words, if I later migrate my website to a different server, will my passwords continue to work?</p>

<p>The method I am using now, as I was told, is dependent on the exact versions of the libraries installed on the server.</p>

If you are choosing an encryption method for your login system then speed is not your friend, Jeff had a to-and-frow with Thomas Ptacek about passwords and the
conclusion
was that you should use the slowest, most secure encryption method you can afford to.
From Thomas Ptacek's blog:
Speed is exactly what you don’t want in a password hash function.
Modern password schemes are attacked with incremental password crackers.
Incremental crackers don’t precalculate all possible cracked passwords. They consider each password hash individually, and they feed their dictionary through the password hash function the same way your PHP login page would. Rainbow table crackers like Ophcrack use space to attack passwords; incremental crackers like John the Ripper, Crack, and LC5 work with time: statistics and compute.
The password attack game is scored in time taken to crack password X. With rainbow tables, that time depends on how big your table needs to be and how fast you can search it. With incremental crackers, the time depends on how fast you can make the password hash function run.
The better you can optimize your password hash function, the faster your password hash function gets, the weaker your scheme is. MD5 and SHA1, even conventional block ciphers like DES, are designed to be fast. MD5, SHA1, and DES are weak password hashes. On modern CPUs, raw crypto building blocks like DES and MD5 can be bitsliced, vectorized, and parallelized to make password searches lightning fast. Game-over FPGA implementations cost only hundreds of dollars.