There are four main ways to gain Twitter followers: build, befriend, beg or buy.
1. Build. When you tweet quality content, people will find you, and follow you. This takes time and consistent effort, however, it means that you will have followers who are interested in your content.
2. Befriend. Find people with similar interests, and "follow" them first. If they see that you are tweeting about a relevant topic, they are likely to follow you back. Engage with other people. Respond to their tweets with an @message.
3. Beg. Or ask, really. Ask your e-mail list, put a your Twitter information on your social networking profiles and a Twitter icon on your web site and the bottom of other online communications. If you use Gmail, Yahoo or AOL for e-mail, you can search your address book for others who are already on Twitter.
4. Buy. Yes, you can buy Twitter followers, but DON'T. If you buy into one of the "gain followers fast" schemes, sure you will boost your follower numbers, but the only value you get will be bragging rights. You'll end up with a large list of people that have no interest in your content, don't click on your links, don't engage with you and just generally don't engage with you/your brand.
Please add your thoughts or experiences about gaining followers in the comments below.