apt-get is a package manager, git is source control. The difference is that apt-get packages are usually pre-built binaries, and they are automatically installed (along with any dependencies). Git is just a place for people to keep their source code, and it tracks each change (in case you need to revert some or all of your source). Git is geared toward software development that involves more than one person (whether it is a commercial team, or random internet people contributing to an open source project).
Github doesn't own Git (many people initially assume that), they just happen to be the most prominent free Git host.
To clone a repository, you just use
git clone <url>
, and a new folder with all the same info and history of changes as the remote repo you are cloning will be created. You can then make changes on the source files, or build the project (how it is built depends on the project).
If you need to update the repo, just go into its folder and hit