A sponsored tweet is an advertising opportunity accepted by a Twitter user. This works the same way as Nike approaching Serena Williams or Go Daddy working with Danica Patrick to promote their products. In sports, the player will wear sponsor logos on clothing or equipment. Online, the sponsored tweet is posted by the Twitter account holder.
On occasion I am offered a sponsored tweet opportunity and I evaluate the merits of each one individually. If I determine the message fits with my brand and my audience, I will accept the sponsorship.
I decline any sponsored tweets when I do not believe in the product or service advertised.
The sponsor usually allows the Twitter member to write their own message using keywords provided. This is one of my recent tweets:
Sponsored: Earn money with Twitter http://spn.tw/r4sD
Each sponsored tweet includes a notice so the reader knows the message is sponsored. Full discloser, these links take you to the Sponsored Tweet site where if you can sign up, I will receive a referrer fee.