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API stands for "application programming interface". In this context is generally means that other people could create applications that could use Twitter's data and functionality.
For example, with the Twitter API, if I had a Hubskier's Twitter username, I could know who they follow on Twitter, and maybe use that data to help make following recommendations here on Hubski. i.e. you follow Bob on Twitter. Bob is here on Hubski, so I'll suggest that you follow him here too. I assume that Twitter would be in the same business, more or less, but make $ my sharing their API, rather than by ads. Eventually, Twitter could fade further into the background, I suppose.