No, Google didn’t buy Twitter. However they launched a similar service called Buzz.
Twitter likely choose to remain a private company because they have plenty to ride on with venture capital and don’t need to sell public options. Leaving the company vulnerable to a hostile takeover wouldn’t be advantageous to them right now; if they sell, they could likely do so for well over whatever their share price would be.
Twitter (contrary to the answers in this thread) are currently receiving revenue from Google and Microsoft as part of the real-time indexing deals they have with Google Search and Bing. (Read more) They also announced paid business accounts, which are currently in a very limited testing phase. (Read more)