Homebrew is a package manager for macOS that allows to easily install third-party software and to keep it up-to-date. If you have ever used Ubuntu, then Homebrew is much like aptitude (apt).
To install Homebrew, open the Terminal and copy & paste the command
$ /usr/bin/ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"
Homebrew installs all executable programmes in /usr/local/bin and this path is not in the default executable path of Mac OS X. To add the path for your user, you can edit the file ~/.profile and add the line
export PATH=/usr/local/bin:$PATH
This will add /usr/local/bin to the PATH variable. You may also consider to add that path to the default environment.
To maintain Homebrew (update the list of available programmes and their versions and update the installed programmes) you may want to place this script under /usr/local/bin. Once the script is place there, you can call it from the Terminal to maintain Homebrew at any time.
Turn off analytics
Turn off analytics (sending anonymous data) by typing
$ brew analytics off