What is a Third Party Cookie?

Let us say for the sake of example that you are surfing Facebook.
Let us say that there is an ad for Joe’s Taxidermy.

If Joe’s Taxidermy tries putting a cookie on your computer, that is a third-party cookie. You never actually went there; they are a third-party in your Facebook surfing. Facebook would be a first-party since you actively went to their site.

Now let us say that you play a Zynga game on FB that uses Flash. Zynga may try to put a cookie on your computer, but since you actually went there, but Adobe would be a third-party; just because you are using Flash, that doesn’t mean that they are actually a primary participant. They are technically a third-party, but like getting usage statistics on their products (in this case, Flash), so third-party cookies are not always a bad thing… though they usually are.

@jerv ‘s answer was probably more accurate but I like to think of them as cold war spies.

Exactly as @jerv said.

But what makes third-party cookies useful to marketing scum advertisers, is that once you have that third-party cookie from Joe’s Taxidermy on your computer, you next go to Answerbag who also happens to have an ad from Joe’s Taxidermy.

Guess what? Joe’s Taxidermy knows what some of your favorite web sites are.

Now replace Joe’s Taxidermy with Google Adsense, who hosts the vast majority of web ads, and they can track your every move on the web.

Any pros and cons of hanging on the cookies on the computer?

You should ditch third party cookies. But most other cookies to sites you use (such as Fluther) are rather useful as they store your preferences and whatnot.