Download and open Minecraft.dmg. Drag Minecraft to your Applications folder to install it. Then run game just like you would any other application. You'll need an internet connection the first time you launch the game, but after that you can play offline without any issues. Browse and download Minecraft Mediafire Projects by the Planet Minecraft community.
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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
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
Minecraft Mediafire Download Zip
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.
Mediafıre Download Minecraft
You'll need an internet connection the first time you launch the game, but after that you can play offline without any issues.