What is a mature PHP solution to use for user logins?
Allow me to recommend OpenID to you. Instead of implementing your own authentication scheme, you delegate authentication to an account provider that your user already has (such as Google, FaceBook, LiveJournal, etc.)
Libraries that implement OpenID on PHP for you include:
There are a lot of different ways to do it depending on use of cookies/database/etc. I have used this php login system before with some luck. The great thing about what you will find is that you can mix and match depending on what you're trying to do. The system I referred also has administration/user levels for varying levels of access. You will find a good tutorial on rolling your own here.
I would suggest reading up using password salts, PHP cookies, and sessions if this is going to be world visible.