Ad Hominem:

This happens when a person talks about the person who brought up the subject, not the subject itself.

Example: "Johnny says the world is round, not flat" "Well, Johnny picks his nose, are you going to believe him?"

Confirmation Bias:

This means ignoring (usually without realizing it) things you see that show that a belief is wrong, while holding onto the things that show you it might be right.

A good example is people who take medicines or treatments that have no science behind them. They often remember all the times it "worked" (when the person got better by themselves) and forget all the times it doesn't.

Straw Man:

(updated thanks to sdavid1726 and nanothief)

This is when someone disagrees with you, so they make an argument like what you said, but not the same thing, in order to make you defend a position other than what you started with.

Example: A parent is trying to get their child to do their homework, and the child says "You just want me to do homework because you don't like me playing games and having fun."

The parent might really think homework is important because it will make their child smarter, but now they will feel like they need to prove that they don't mind children having fun, so they've been distracted from their original meaning.

Appeal to Authority:

This happens when a person says that something is right just because some important person says its right.

Example: "I'm not going to give my daughter the injections the doctors say she should have, because Jenny McCarthy says they're bad."

Appeal to popularity:

If you hear someone say that something is right, because lots of other people think it as well, this is "appealing to popularity".

Example: Hearing your friends say "I should get my ears pierced, because all my friends have their ears pierced."

Slippery Slope:

Sometimes you hear people say that if one thing happens, then a lot of other things must follow, and soon something awful will happen.

Example: At the moment, a lot of people are arguing about whether men should be able to marry other men, or women should be able to marry women. I've heard some people say that this is bad because if we let this happen, then soon brothers and sisters will be allowed to marry and even that people will be allowed to marry dogs. People who say this are making a slippery slope argument.

EDIT:

Changed Straw Man to include nanothief's better example.

No True Scotsman:

This usually happens when someone thinks "their kind of people" would never do anything wrong. When they are shown otherwise, they will try to remove that person from "their kind of people." To some extent, this is linked with Confirmation Bias, because whoever does this is trying to ignore the evidence that doesn't support what they already believe.

Example:

There have been a series of kids caught cheating in their tests in the schools around the city. One school principal says "That will never happen at my school. My kids are too good to ever do that." The next week, one of his kids is caught cheating. Faced with this news, the principal says "Ok, but Johnny only started here three months ago, he's not really one of us, yet."