What is the correct name for posts made on twitter?

Twitter is a brand and, as such, you should follow the usage established by the brand itself. Looking at http://blog.twitter.com/, I see they describe the messages as Tweets (singular: a Tweet), with a capital T, and the action of sending them as tweeting (verb: to tweet), no capitalization.

That rule should be followed until established usage starts to differ from what the brand proposes, either because people think it's silly (like writing Yahoo! with its exclamation mark all the time) or because they think it's ugly (I would personally not use a capital T for tweets, as it disrupts the standard rules of capitalization).

To tweet a tweet

Twitter.com prefers the noun Tweet to be capitalised but dictionaries have it without the capital T

• Please tweet this to your followers
• I was tweeting about Twitter yesterday

Oxford dictionaries:

tweet:

A posting made on the social media website Twitter: he started posting tweets via his mobile phone to let his parents know he was safe _{Oxford Dictionaries.com}

Twit:

A silly or foolish person. _{Oxford Dictionaries.com}

Mostly used in British English

It also depends on the level of formality of your writing. In the summer of 2010, the standards editor for the New York Times declared the policy for the NYT:

Except for special effect, we try to avoid colloquialisms, neologisms and jargon. And "tweet" – as a noun or a verb, referring to messages on Twitter – is all three. Yet it has appeared 18 times in articles in the past month, in a range of sections.

...

"Tweet" may be acceptable occasionally for special effect. But let's look for deft, English alternatives: use Twitter, post to or on Twitter, write on Twitter, a Twitter message, a Twitter update. Or, once you've established that Twitter is the medium, simply use "say" or "write."