Types of Competitions
The most popular type (and cheapest to run) competition is the simple prize draw (e.g Loquax Prize Draws). These often require a user to submit their details and perhaps answer a simple question (based on the visible promotional text or about a sponsor or about part of your website).
A treasure hunt is a good way to entice people into looking all around your site. They can be used to encourage visitors to look at new or under used parts of your site. Games or quizzes can also be popular, but take time to develop and can be costly.
If you're looking for some initiative from your visitors, why not make them use their imagination by writing a slogan/tiebreaker (e.g Loquax Tiebreakers), a caption or limerick on a theme or picture related to your business?
For more creativity consider a photo competition or even a movie competition using uploads to YouTube? Social Networking has become popular and some competitions are now run based on getting people to join a group on Facebook or Bebo.
Whatever type of competition you decide to choose, make sure that you choose something that will be fun for your visitors whilst being easy for you to maintain. For example, slogan/tiebreaker competitions will require judging, whilst long quizzes may require someone to check all the answers
The key is to run a competition that best suits your aims and your website..