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.