OpenID is a way of using one ID for multiple sites. Think of it like using a Facebook login to register on multiple sites, just without the Facebook account.
The great thing about OpenID is that it's based on secure principles, and you only have to remember one password to identify yourself to any site that chooses to implement OpenID.
The problem, however, is that OpenID relies on OID providers. If you want to maintain your own identity, you're pretty much out of luck. You still have to hand over some of your identity to a third party who maintains your identity to serve it to any website which you authorize to use your identity to log you in.
PHPMyID allows you to essentially be your own identity provider. Instead of registering on one centralized provider to maintain your profile, you set up the script and run it on your own server. This way, you retain control of your own identity, while still using one login for any website that supports OpenID.
Note, of course, that PHPMyID is no longer developed.