You will need Java to play Minecraft. If you don't have it, go get it.

If the files show up as a compressed archive (Zip/Rar) to you, or you cannot open the file by double clicking on it, first ensure you have Java installed. If you have Java installed then try using Jarfix to fix the file association.

Mineshafter Launcher

Login with whatever username or password you want. Servers should ask you to login again once you connect. Please note that you cannot connect to official servers using this. Only open servers, or those using authentication plugins will work.

If you get a corrupted jar error, try getting it from one of these mirrors instead:

MEGA mirror

Torrent (your torrent client must support magnet links and DHT)