Skilled debating is an art. In order to win arguments and convince others of your views, you must understand the basic components of logic, psychology, and effective communication.

[edit] Steps

Decide on a position you would like to argue for, and become well-informed about that position. Ideally, this will be something you really believe in, because it is much easier to make convincing arguments for ideas you are enthusiastic about. Make sure you understand not only your own position, but the opposing position as well. This will allow you to anticipate objections and respond more effectively.
Find someone to debate with. If you are reading this article, it's likely that you already have someone in mind. Before proceeding, however, you should familiarize yourself with the concept of "impossible people." In order to have any chance of winning a debate or accomplishing anything productive, you need to be arguing with someone who is basically reasonable. Otherwise, save yourself the trouble and find someone more reasonable to debate with.
Begin by stating a thesis. This is just a brief statement of your position and your reasons for holding that position. Example: You might say "I believe the Moon was once of a part of the Earth for the following reasons, followed by a quick summary of why you believe that. Try to use evidence-based premises, if possible. For example, "Geological data shows that the Moon's rocks are quite similar to those found in Earth's early history" is much better than "The Moon being blasted out into space by a collision is just a really cool idea.
Respond to objections. In most cases, your opponent will respond to your thesis by objecting to one or more of your premises, which are the reasons you have given to support your position. If you are well-informed about your position, most of the objections should already be familiar to you. Use logic and evidence to show your opponent why his or her objections do not work. You can refute (dispose of) objections by two major routes: showing that the evidence does not support them, or exposing a logical flaw in the premise of the objection. To refute the idea that refined white bread is healthy because it is processed, you might state that a study of rats fed a diet of white bread alone all died (which is true, incidentally). This would be an evidence-based response.
To make a logic-based response, you might state that "The fact that white bread is processed does not mean it is healthy. There is no established link between highly processed food and better health, so your objection does not follow from your premises."
Build on your opponent's objections. If possible, don't stop at refuting them - turn them around and use them against your opponent's position.
- Example: Your thesis might be that lab rats should not be used in painful experiments. Your opponent might object that rats cannot experience pain in the same way humans can. You might use evidence to refute this objection by referring to studies which show the same type of brain and nervous system function in rats and humans under stress. Instead of stopping there, show your opponent how his or her attempted objection actually supports your position. Continuing the example given here, you might say something like "since you have made the issue of the ability to feel pain the basis of your objection, doesn't the evidence I've shown you suggest that performing experiments on lab animals is unethical?"
Attempt to resolve each point before moving ahead to the next issue. If there are unresolved points about which you and your opponent cannot agree, it will be difficult to accomplish anything productive, because the unresolved points will continue to come up over and over again. Ultimately, this will lead to a situation where there is no choice but to "agree to disagree," which is usually not an ideal outcome.
Remain calm, rational, and reasonable at all times. You may feel that your opponent is totally failing to understand your position, but if you become too agitated, you opponent will take this as a sign of weakness and conclude that he has you on the ropes. Rather than helping to convince your opponent, shouting or insulting remarks will only serve to make him more confident in his position. Emotional behavior is no substitute for rational arguments.
Have patience. As long as both you and your opponent are debating in a reasonable manner, be willing to spend some time explaining your position and your premises. It is not easy to change someone else's mind. There are a variety of reasons for this, but the most powerful among them is the simple fact that no one enjoys discovering that he is mistaken. It's not a particularly easy thing to accept, so be patient. You won't convince him with your very first point.
Use effective speech and grammar. You don't need to pretend you are a university professor, but if you want to be effective and convincing, you should use decent English. Don't try to use big words in order to sound more intelligent, because most people can see through such an act. On the other hand, don't be afraid to use the right word for the task. If a big word is called for, use it. Most importantly, try to speak (or write) clearly and confidently. Make your point using no more and no fewer words than you need.
Ask questions. Most people assume that the person with the most knowledge of a topic will win in a debate, this, however, is not true. If you are able to ask questions you can easily even any playing field. The idea behind this method goes back to Socrates. Socrates would ask the men, who thought of themselves as wise, question after question until they could no longer give a response without demonstrating a logical fallacy or proving his point. Remember that many people like to hear themselves talk, this can be used against them. Also, do not use questions that they can have multiple answers for, if they respond with an "um...(pause)" and contemplate the idea to make up their mind you will go nowhere because once you have completed the questioning period all they need to do to avoid your conclusion is to go back to that question and change their mind.
- Using the debate example previously mentioned (the pain-response of rats) a way to debate using the Socratic Method would be to ask "How do humans feel pain." The logical answer would be through nervous-system impulses (you will most likely get an answer much simpler but with that basic idea). You then might ask if a nervous system is responsible for those impulses. They will answer yes, and then you ask if rats have a nervous system. The logical conclusion is yes. Therefore, if rats have a nervous system and a nervous system is responsible for pain, rats can feel pain.
- Another method to argue that same point is to ask how you know someone else is feeling pain. They are likely to respond that the person will say ouch. You then ask them "Well, babies don't say ouch, does that mean that babies don't feel pain?" They'll most likely change their answer to be a bit broader (always try to get them to concede the broadest definition for an idea (i.e. murder, life, pain), this allows for you to make your point part of that definition). They will most likely retract their previous statement and say that if a person cries out then they are in pain. You then point out that rats squeal and try to escape when they are most likely in pain.
Be willing to lose. A skilled dialectician (debater or arguer) understands that sometimes, the other person's arguments will simply be stronger than one's own. If you find yourself cornered and unable to refute an opponent's points, be honest and reasonable enough to concede defeat. Do not become stubbornly determined to keep objecting even after you have been proven wrong (that would make you an "impossible person!"). Anyone who has engaged in debate regularly has experienced a number of losses. Congratulate your opponent, learn from your mistakes, and move on. Every experience (win or lose) makes you better equipped for your next encounter.
Don't take too much pride in being right (or winning the debate). It makes it harder for your adversary to admit to being wrong which, in an informal clash of opinions, should be your primary goal.

[edit] Tips

The rule of thumb to remember is that your own desire to believe something or your own opinion about how great the belief is will not convince anyone else. To do that, you need logic and evidence.
Become familiar with how logic works. Being able to argue logically will make you very effective at pointing out the flaws and contradictions in your opponent's objections. You don't have to take a formal logic course or memorize logical notation, as long as you understand the basic principles of logic and the logical fallacies (arguments that have a logical flaw which makes them inconsistent or self-contradictory). An introduction to logical fallacies is linked below.
On the other hand, if your opponent is an amateur, consider using logical fallacies (see below) of your own, like Straw Man arguments or non-sequiturs.
Be a gracious loser. Never have bad sportsmanship.
Increase your vocabulary to include great segue words and debate jargon to use in your argument, such as "moreover", "inasmuch", "per se", and "vis-a-vis".
If all else fails, remember that you can "win" an argument by just proving them wrong. You don't have to prove yourself right, but since they are wrong this implies you are right.
A well defined technique to discredit your opponents points is a three step process of cusion, question, and confirm. First accept your opponents viewpoint, then ask a general leading question and then confirm that the answer to that question disproves their point. The cusion step is critical to letting your opponent know you understand their point, but disapprove.
Keep talking about facts and logic and such. When your opponent gets bored of this argument, likely they'll apologize and say you were right the whole time. Note: This will take a lot of knowledge on the subject and patience.
Thank the opposite team win or lose, it makes them happy.:)
I advise you to, before your speech, practice gargling, anything to help your speech.

[edit] Warnings

Be careful how passionately you approach an issue. If you ostracize your opponent they will rarely end up agreeing with you, no matter how convincing your point was.
Remember that you cannot win a debate against a friend. If you make them feel foolish you may end up losing their friendship.
Beware as you may stuff up badly, so practice a lot! I recommend to practice in front of a mirror or someone who will listen and criticise you.
If you're in an informal argument, your opponent may resort to one-liners like "You have no life". They may also attempt to threaten or shout at you. All of these things, however, are merely de facto examples of people who have been defeated in an argument.