An application, as far as Twitter sees it, is just a row in a database on our servers. An application has an API key, an API secret, and users who have authorized it for use. “UberSocial for BlackBerry” is an application in our database, and every phone that runs this app uses the same API key – it’s all tied to the entry in our database representing that application.
Outside of Twitter, an application has a much broader meaning. Some Applications are distributed without their own API keys – in this case, you as a user must obtain your own API key from Twitter by registering “an application” (as Twitter sees it, again, a row in a database). One of the rules of registering an application with Twitter is that the name needs to be unique.
Hope this clarifies that for you.