What is the correct name for posts made on twitter?

Solution 1:

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

Solution 2:

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).