1. The Entry with the Most Votes Wins
This is the simplest method of picking a contest winner: the entry with the most votes win. The advantage for sponsors is that it's easy for them, and there's no risk that they'll be accused of playing favorites since they're not the ones deciding the result. For sponsors, there's also a risk that the winner won't be the entry that they like best. In other words, it might not be the best fit for their image, their advertising campaign, or their contest goals.
There's a disadvantage for entrants as well. Oftentimes, votes don't go to the "best" entry but to friends or family members. When a contest boils down to popularity, there's a risk that someone with a mediocre entry can still generate tons of votes overnight. It's frustrating to lose to a weaker entry that was made by someone with broader reach.
2. Voting Selects Finalists, Judges Pick Winner
In this type of contest, the results of the vote select a given number of finalists, and then the judges pick the winner from among the finalists. This has an advantage for the sponsor of not having to comb through every single entry (which can be very time consuming and expensive), while still allowing the sponsor the control to make sure they approve of the final winners.
Entrants don't have to worry quite as much about the popularity factor in contests, especially when there's a high number of finalists to be selected.
3. Judges Select Finalists, Voting Selects Winner
Contests can also select winners the other way around - the judges select a given number of finalists, and the vote picks who'll win from among the finalists. For sponsors, this means that they have to sort through all of the entries, but the advantage is that all of the finalists will be acceptable winners.
For entrants, this means that all of the winners should be strong entries. Although it's still possible that one of the finalists will have a lot of "pull" and be able to draw a lot of votes, merit has a lot more influence on the winner than a pure voting contest.
Note: It's possible to combine voting methods; for example, a contest might have judges select a bunch of semi-finalists, a public vote pick some of those entries to be finalists, and then have judges pick the winner from among the finalists.
4. Vote Results are One Factor in Selecting Winner
Probably my favorite way of using votes to influence the outcome of a contest, this method uses a variety of factors, including voting, to determine the winner. For example, the sponsor might assign points for getting a lot of votes as well as for other qualities they're looking for in their entries, like relevance to a theme, creativity, and/or emotional impact.
I like this method because it has a good chance of picking the "best" entry as the winner, while reducing the impact of popularity (or cheating!) on the outcome.
5. A Separate Prize is Awarded for Getting Votes
In this method, the sponsor awards a separate prize, often called a "Voter's Choice" or "People's Choice" prize, to the entrants that receive the highest number of votes. This has the advantage that the judges have complete control over who wins the main prizes, but the contest still attracts traffic to the sponsor's website from the voting element.For entrants, this type of contest offers two separate ways of winning, giving you the opportunity to win prizes through the popularity of your entry and based on how much the judges like it.
6. Votes Don't Influence the Winner of the Contest
Of course, not every contest uses voting at all. Voting has an advantage to the sponsor, because it brings more people to their website and builds "buzz" about their contest, while giving the public a say in who wins. But on the other hand, a contest that uses votes needs to be coded robustly to avoid accusations of cheating.
For contestants, it can be fun to tell friends and family members about their contest entries, but it can also be stressful, especially if another contestant is having more luck drawing votes.
7. Conclusion
How a contest uses votes to pick the winners can make a big difference on the outcome, and on how difficult a contest is to run. It makes sense for entrants to evaluate how "fair" the results are likely to be when deciding how much effort to put into your contest entries.

