My understanding of BitBucket is that it is like GitHub, but does not use GitHub. However, it does use Git.
Git is version control software. BitBucket uses Git to make commits and branches and diffs and all that, but it adds a web interface. It is also, as you say, a source repository. It provides ways to share this repository in the cloud to multiple programmers.
Eclipse is, as you say, an IDE. I don't know of any integration of Eclipse and BitBucket.
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