OpenStack is an open-sourced, IaaS cloud computing platform founded by Rackspace Hosting and NASA, and used widely in industry. Some of the modules that it provides to users includes compute, storage and network components allowing users to get easily started in cloud computing.
Many clouds exist that have OpenStack as a foundation as part of the cloud services offered to users. Some of them are free, some are commercial services.
Indiana University maintains some compute resources that you can use for classes.