To install a package (or Formula in Homebrew vocabulary) simply type:
brew install <formula>
To update Homebrew's directory of formulae, run:
brew update
To see if any of your formulas need to be updated:
brew outdated
To update a formula:
brew upgrade <formula>
Homebrew keeps older versions of formulas installed on your system, in case you want to roll back to an older version. That is rarely necessary, so you can do some cleanup to get rid of those old versions:
brew cleanup
If you want to see what formulae Homebrew would delete without actually deleting them, you can run:
brew cleanup --dry-run
To see what you have installed (with their version numbers):