You press check by pulling a handgun's slide back just enough to see if a round is chambered. You check so that you know your handgun is prepared. The last thing you want to occur is to be attacked but find that your gun doesn't have a round chambered.
Some handguns have a device called a loaded chamber indicator. It's a small tab that rises when there's a round in the chamber. This can be used in lieu of a press check, but not all handguns have LCIs. Some handguns have external extractors that pop out when a round is chambered, but not all handguns have external extractors. As well, not all extractors pop out enough for the wielder of the gun to know that a round is chambered. And lastly, not everyone is willing to trust an LCI when a visual verification works better. I'm one of those people.
Here's what happens if you don't press check (watch the video):
The video is captioned:
People like give me shit because I press check my guns (I.e. Partially pull the slide back to see if there is actually a round in the chamber. Well, This is why I press check. I get that it's all the rage now the make fun of things that look high speed low drag but sometimes there is purpose. #IPressCheckedButDidYouDie
The above is a training exercise but highlights the fact that if someone is defending themselves but don't know the condition of their gun, they can die. No one wants to hear "click" when pulling the trigger in a dire situation.
Another reason to press check is that sometimes magazines might not be properly seated, so that when a slide is racked, a round is not stripped from the magazine and pushed into the chamber. In that case, you'd think that a round was chambered, only it never did. An LCI would indicate that the chamber is empty, but a press check would do the same and not rely on another gun part (that could possibly fail).
Another good source of information is The Firearms Blog. They published an article about their thoughts on it today, and they pretty much mimic Colion.
There's nothing really wrong with press checking. If you're doing it every 60 seconds on a firing line at the local range, you're doing it wrong, though. The practical applications outweigh the tacticool-ness of press checking.