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>.