Are you referring to the EC2 platform? You do have to pay for that unless you fall into the "free usage" tier.
I've set up EC2 instances in the past, and personally, I don't think it's worth the trouble. Quality web hosting can be easily found for $5/month or so, which is absolutely worth it to me for the simplicity and support. Running your own EC2 instance creates some technological overhead that you'll be responsible for; there will be no support if you screw something up.
Amazon wrote a good how-to for setting up a web server on EC2 using their custom Linux distro. Take a look and see if you feel comfortable enough to do it:
Tutorial: Installing a LAMP Web Server on Amazon Linux - Amazon Elastic Compute Cloud