I can only play Minecraft demo
If you would like to try Minecraft before deciding to make a purchase, you can play the demo mode by registering for an account, then downloading the Minecraft launcher. This version of the game lasts five in-game days, or about 100 minutes. You are not able to connect to multiplayer servers in demo mode, but you are able to play over LAN. The demo mode is available to any Mojang account that is logged into the launcher but does not own a copy of Minecraft.
If you have purchased the game but for some reason, you still see DEMO MODE, make sure to contact official Minecraft support team.
How can i play for free?
If you would like to play for free there are many different launchers around the web. You can find few of them in "Launchers" section of this page. (If you do so, notice that these launchers are meant for testing and developing custom functions. As soon as you can, purchase full game as it offers access to premium servers, official skin system and you are supporting amazing developers.)