When I shorten a link via bit.ly or something like that, I usually do not see any clicks on it (even if the tweet appears in Google). Does it mean that Google ignores them?
They are no good for backlinks to build your search results, but if you have a decent following with people who will actually click those links and view your page, then your site views will increase and therefore it could move up your search results. People visiting your site is more powerful then just accumulating a bunch of backlinks, so it could possibly be even more effective then getting a random backlink somewhere.
Tweets are nofollow so I think it doesn't have effects for backlinks but as jwarnimo said you can get traffic from your followers.
Twitter links couldn't be called backlinks, but they have a value. Social engines like Twitter, Facebook or Google Plus have very large value in Google Algorithms.
twitter links are good even it is nofollow but it still beneficial for driving traffics to your site.
As others have said, twitter links are nofollowed so they probably don't have a direct impact on rankings. URL shorteners like bit.ly use a 301 redirect, which transfers almost all of the ranking power, so using shortened links should have little to no effect on how much pagerank is passed, but the nofollow means that no pagerank is passed.