Can I download minecraft on different computers with 1 account?

I want to download minecraft on 2 different computers without having to buy it twice. I have logged in and tried it get it, but it won't let me! It will make me pay!

Yes you can. When you buy the game you are entitled to unlimited downloads. Simply go to the new computer, sign in to minecraft.net and you may re-download from there. However, any saves you had on the original computer will not carry over. To do that you'll have to locate the save files and copy them to a thumb drive or something to be transferred. There are plenty of videos on YouTube about finding the save files.