DEBIAN AND DEBIAN BASED DISTRIBUTIONS

If you use a Debian based distribution (like Ubuntu), download the .deb package and install it. This should install all the required dependencies for you. If your environment doesn't have a graphical package installer, you can install the package from the terminal, like this:

apt-get install Minecraft.deb

ARCH AND ARCH BASED DISTRIBUTIONS

In your Minecraft server folder, look for the file named 'server.properties' and open it with your favorite text editor. The Minecraft wiki has an exhaustive guide for configuring your server.

On Arch and Arch based distributions, the package is available as minecraft-launcher from the AUR. AUR is a repository of packages maintained by the community. You can read more about how to use it on the Arch wiki.

OTHER LINUX DISTRIBUTIONS

On other distributions, you can download just the raw launcher files as an archive, extract them somewhere and run them in any way that is convenient (from a terminal, creating a shortcut, etc.). You will need Java, and some dependencies, most of which should already be present on common linux desktops. This is definitely the 'some assembly required' option though.

You'll need an internet connection the first time you launch the game, but after that you can play offline without any issues.