Check that the argument is supported by facts. When people use facts in an argument, it lets you know that they are drawing their conclusions from provable concepts, rather than opinions or assumptions. By definition, facts have to be supported by concrete evidence. When people use facts, they should cite their sources so you can look them up and learn more. If you are unsure about a fact or don't know its source, research to see if it's valid.
Watch for circular reasoning. If the conclusion sounds like a restatement of all or part of the argument itself, the argument is invalid, because arguments cannot prove themselves. For example, "Carrots are nutritious because they are good for you," is not a valid argument, because "carrots are nutritious" and "carrots are good for you" are the same idea. Most people know that carrots are healthy, but you cannot validly argue that they are without introducing new information.
Check to see if opposing arguments are being addressed instead of avoided. For every argument, there is always a counterargument. If the person shies away from counterarguments or tries to steer the conversation in another direction, this could be a sign that he doesn't have enough facts to back up his conclusions or his argument contains a fallacy. When people switch topics midway through an argument or try to support an argument with irrelevant facts, they are employing a red herring.
Make sure the opponent's viewpoint is not being oversimplified. When people create an argument around a hollow viewpoint, it is called the straw man fallacy. For example, the argument "People who don't eat carrots obviously don't care about their bodies," makes a gross generalization about why people would choose not to eat carrots.
Watch for arguments that jump to conclusions. Proving that something could happen does not necessarily prove that it will happen. This is called the slippery slope fallacy. An example would be, "If more people start eating carrots, the demand for french fries will drop so much that fast food restaurants will go out of business." Although this scenario is possible, it is both unlikely and unprovable. The argument falls apart because it assumes that people who eat carrots won't eat french fries and fast food restaurants cannot adapt their menus to stay in business.