"Amazon EC2’s simple web service interface allows you to obtain and configure capacity with minimal friction. It provides you with complete control of your computing resources and lets you run on Amazon’s proven computing environment. Amazon EC2 reduces the time required to obtain and boot new server instances to minutes, allowing you to quickly scale capacity, both up and down, as your computing requirements change. " by Amazon EC2.
I just see the introduction about Amazon Elastic Compute Cloud, Amazon EC2.
It is a web service that provides a running environment and compute capacity in the cloud, which means in their server.
Their website also provides a detailed information about Amazon EC2. You can see the Amazon EC2 Functionality, Service Highlights, Instance Types...etc on this website.