There are multiple ways to handle a Twitter giveaway. I thought it would be a good idea to put together a helpful guide to giving stuff away on Twitter. People have the best intentions when doing this, but a lot of times get tripped up by some of the lesser known features of Twitter. You should always be well prepared before you send a contest tweet in order minimize frustration on both ends of the giveaway.
A lot of people prefer to post codes that can be redeemed on Xbox Live or iTunes. You may want to just post the code as is, or make it a little more difficult and change one or more of the characters to question marks and have people guess the right letter or number. This can make it more interesting, but keep in mind that not everyone is on an even playing field in these situations. Different browsers and operating systems can lend advantages or disadvantages to the copying and pasting of codes. Most people won't complain because it's great that you're giving away some swag. Just keep in mind that it can be a little frustrating and the point of a giveaway is to make people like you. It's also nice to drop a warning tweet 5 to 10 minutes before you actually post the code.
Another way is to let people enter your contest by direct messaging you. You can ask for a DM with a specific word, phrase, or even the answer to a trivia question. Then you can choose an entrant at random or pick the first correct entry. Getting DM's takes a lot of the frustration and guesswork out of a giveaway because only you can see the entries. Once chosen, simply DM the winner with information on their prize and then let the rest of your followers know who won.
One problem with the DM method is that only people you are following can send you a direct message. Unless you follow everyone who follows you, this can be a pain. If you're running a community Twitter account with a lot of followers, make sure you follow everyone back. Otherwise you will try to hold a contest and people will not be able to DM you. Then you can expect a flurry of angry tweets flooding your @ box. This is probably the number one mistake people make simply because they don't realize not everyone can DM you. Twitter should really enable a check box on your settings so you can allow DM's from everyone.
The retweet contest can work if you plan on getting entries over several days or even weeks. Simply create a post on your site with the text you want retweeted. Then tweet the message with the link to your post and your followers will do the rest. This method can both drive traffic to your site and bring you more followers. Retweeting is also a really easy way to enter for people. You will just need to make sure to specify how many entries are allowed per day. Use Random.org to pick a winner and then bask in your own awesomeness. A good example of a site that uses the retweet method is Week In Rewind (http://www.weekinrewind.com/search/label/Contests).
One thing to be mindful of on the retweet. The character limit is 140, so you will need to leave enough room for people to include the "RT" and your @name. That way you don't have people editing the tweet in order for it to fit.
There are other methods to drop free stuff onto your followers, but these seem to be the most popular. Incorporating your website in some way is also a plus. Have your followers find a specific piece of information from your site and DM it to you. Use Twitter to drive them to a post and have them comment on that post. For you podcasters, ask for a review in iTunes to act as the entry. You could even have people listen to a particular episode and have them DM or e-mail in a specific word or phrase you give out. The possibilities are practically endless.
I want to thank @seeson from 360Junkies and @Steve519 from PlatformNation for their input on this!