Basically AWS S3 is a file system. It acts as network disk. If you have no prior knowledge of cloud, you can treat it as dropbox. AWS Storage Gateway is an virtual interface which enables you to read/write data from/to aws S3 or other aws storage services.
You can relate S3 as dropbox itself, that you can access through web or api, and AWS Storage Gateway as the dropbox client on your pc, which simulate the dropbox as your local drive .