The Application Programming Interface, API, is a large part of Twitter's success. Twitter allows developers around the world to have free and open access to see under-the-hood of how it works. Think of the API as a thin, see-through layer that sits on top of Twitter and can pull through any of its raw material, like incoming tweet data from around the world, and use it in new and exciting ways.
Anyone with free time who loves software coding can write applications, like Hootsuite, to use Twitter better, or differently, or in combination with other social media platforms, or to collect metrics on how an account performs or how a topic trends. Twitter has every right to deny you access though, so tread carefully, get an OAuth access token and read all of Twitter's API terms.