I <3 Twitter, but I still don't understand the retweet. Do you just hit the "reply arrow" next to the person's tweet and then hit capital RT and then type your tweet?
— Troubled in Twitterland
What you're describing is a reply. When you're at Twitter.com and rollover someone's tweet, you'll see the arrow. By clicking on that icon you'll be replying to that person, not retweeting them. (Retweeting, or RT, is repeating what they said while giving them credit.)
Other 3rd party apps, like TweetDeck, have a similar method of replying to someone, using that curved arrow:
Twitter.com doesn't have a default retweet feature, but the most widely accepted RT format is "RT @twitterhandle:" and then their tweet. Another option is just appending (via @twitterhandle) to the end of their tweet; however, it adds 7 characters to the RT as opposed to 4 characters in the first example.
If you're using TweetDeck, rollover your tweep's avatar and choose the RT arrow, which is the non-curved one:
If you're looking to get your own tweets retweeted, it's helpful to know your "magic number", or the number of characters someone will use to retweet you subtracted from 140. So, if "RT @therichbrooks: " (don't forget the spaces) takes up 19 characters, my original tweet needs to be 121 characters or less.