To “brick your phone” is to render it useless (like a piece of brick). It probably is a play on the phrase “break your phone”. Because essentially when you “brick your phone”, you have broken the software that make the phone runs.
This usually happens when one attempting to do some hack on an Android-based phone. There are many articles on the web with instructions on how to “hack your phone” to get it to do things that it normally would not do. For example, lifehacker.com has an article on how to tether your Android phone.
Or it can happens if one runs a misbehaved apps on a smartphone.
If you use your phone as a phone, or use it as it should be used, then the chances of bricking your phone is quite slim.