There are several things going on with a tweet link.
The actual href is always replaced with a t.co URL, to help you fit into the 140 characters (even if you've got plenty of characters left).
The link has rel="nofollow" on it, so search engines are not necessarily going to pay it a whole load of attention - it doesn't really matter that your own URL isn't there.
There is an attribute called data-expanded-url added to the link, with the full and genuine URL as the value, so any robots who are interested can get the genuine URL without having to go through the t.co redirect.
If your URL is short enough (28 chars?) it is displayed intact in the tweet, otherwise it is truncated.
I do have to laugh the way so many tweeters go to the effort of shortening their URLs on bit.ly, ow.ly or many others when they could just bung the whole URL in there and let Twitter sort it all out. Still, I suppose it hides the fact that they're usually using stupid ugly long URLs that they shouldn't have in the first place ... but then I like to laugh at any organisation that uses a shortened URL in any of its official publicity ... have they not heard of 301 redirects?