brew
- core command for Homebrew. Typically deals with command line software.
brew cask
- extension to brew that allows management of GUI applications.
formulae
- package definition files for brew.
cask
- package definition files for brew cask.
(casks and formulae are written in a Ruby-based DSL that describe how to install something)
Cellar
- where Homebrew install things on your system and then adds symlinks from standard locations to it.
tap
- source of formulae. Default is homebrew/core
. You can add more, and even your own by creating a github repo called homebrew-<something>
, putting your formula file in it, and then using brew tap <username>/<something>
.