The installation is quite simple as there are seperate installers for MacOS and Windows machines available here. Currently two versions are for download: one beta and one stable.
Setup will start after you downloaded the program and you´ll need to login with your GitHub credentials. That is really the only step because after that you can start to create a repository or clone one.
Note: during the installation not only GitHub Desktop will be installed but Git too. So you don´t need to install it seperate.