What does "to root a phone" mean? I heard this phrase or the variants (rooted phone, rooting, etc.) in android forums but I don't know what it means. Can you explain it in simple terms?
To 'root' your phone means to gain administrative rights on the file system of your phone (in linux, root is the username of the master admin).  With root access, you can install and uninstall anything you want on the phone.

Most phones come with limited access regarding what you can and can't do on it.  This isn't necessarily a bad thing, because it can keep users from accidentally breaking something they shouldn't mess with on the phone, especially in regards to the operating system.  However, many manufacturers limit your rights to things that aren't really so mission critical, too, and rooting the phone gets around this.

If you have a few unnecessary applications (bloatware) pre-installed on your phone that you cannot uninstall, rooting will give you this ability.  It will also allow you to upgrade to newer versions of Android before your phone's manufacturer and/or cell provider make the updates available to you.  Be warned, though, that rooting may void your warranty.

For more info on what rooting enables you to do with your phone, check out this question.