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>.