In this article I want to explain what Amazon's EC2 service is. This is targeted at lazy people like you who do not want to read the whole Amazon documentation to get started:

What is EC2?

EC2 is short for "Elastic Computing Cloud" and basically provides its users with the ability to boot up Virtual Machines (VM) of every shape and size.

What can I use it for?

The possibilities are almost endless. Most of the time it is used for booting up servers and hosting some kind of website or application.

How can I use it?

First, you have to have an account at AWS. You can get one here. There is a free tier, so you don't pay for just playing around a bit. A credit card is needed nevertheless.
Then you can log into the AWS Console (that's what they call the web application you can manage the services from).
Here you click "EC2" to get to the EC2 Dashboard. You should have arrived in the same virtual machine wonderland like me:

How can I boot one of these "VNs"?

First of all, they are called "VMs". Please don't embarass us in front of the other nerds!
Here is what you do:
Click on Launch Instance in the EC2 Dashboard

Now you can choose an AMI and get going with the...

Wait! What's an AMI?

The AMI or "Amazon Machine Image" contains (among other stuff) the OS you want to boot. At this point you will have to choose what OS your VM is going to have. As you can see, there are already preconfigured AMIs for Linux and Windows. You can either just grab one of these and get going or head over to AWS's marketplace and get a special AMI. Here you can find AMIs which contain solutions for hosting CMS like Wordpress or Joomla!. If you are not so experienced in configuring servers, I defenitely reccomend this approach.

Ok, but which AMI do I choose?

Let's keep it simple for the sake of reader's lazyiness:
Choose the Amazon Linux AMI. This version of Linux is specially configured to work together with AWS. Click on the select button:

Now you will get to choose an Instance Type!

What? I thought I just chose Amazon Linux as an Instance Type?

No, you didn't. The Amazon Linux was the AMI. The Instance Type is the hardware your VM is running on.
You can go really crazy here. Depending on your computing power needs, you can choose from micro instances up to 10xlarge instances. Notice how I didn't do a Yo Mama joke on the last one.
If you want to use AWS free tier you will have to use the smallest one:

The more computing power you choose the higher your bill is going to be at the end of the month. If you are not sure how much you are going to pay you can use the rediculosly ugly cost calulator here.
Click on "Review and Launch", act like you are reviewing your set up and launch it.
You will then have to create a new keypair and download it:

You will later use this keypair to access your VM via ssh.

Congrats! You launched a VM, you badass!

How can I access my machine?

You will need to have a ssh tool like putty to get access your machine via ssh!
Before you can do this, you have to convert the key's format into the .ppk format.
If you do not know how this or sshing works, you should follow this tutorial from Amazon:

Now you have a working Virtual Machine in the Amazon Cloud - and you can do anything you want with it!