To install a package (or Formula in Homebrew vocabulary) simply type:
$ brew install <formula>
To update Homebrew's directory of formulae, run:
$ brew update
Note: I've seen that command fail sometimes because of a bug. If that ever happens, run the following (when you have Git installed):
$ cd /usr/local/Homebrew/
$ git fetch origin
$ git reset --hard origin/master
To see if any of your packages need to be updated:
$ brew outdated
To update a package:
$ brew upgrade <formula>
Homebrew keeps older versions of packages installed, in case you want to roll back. That rarely is necessary, so you can do some cleanup to get rid of those old versions:
$ brew cleanup
To see what you have installed (with their version numbers):
$ brew list --versions