How do I get the cork out of the bottle?

Sorry,

Well, there's actually 2 answer for that, the wine expert in me what to say - don't bother, the cork oxidize the wine and ruins it and a wine that a cork fell into should just go to the garbage and not be drank at all. Yes, cork does ruin the wine. However, since at times I just want to drink some wine and don't really notice all the subtleties of flavor and smells and just want to drink it - it's possible to drink the wine with the cork still inside the bottle - the flavor will change but it's still drinkable and wouldn't risk your health or anything, there's really no need to take the cork out of the bottle. If you are serving the wine to unsuspicious guests and don't want them to know you messed up by dropping the cork into the bottle - just pour the wine into a carafe leaving the cork in the bottle.

I once heard that it is possible to remove the cork from inside the bottle by looping a piece if cotton thread around it and pulling it through the neck of the bottle. I've never tried it, and it seems to me that it would be extremely difficult to get the cotton thread and the cork aligned properly with the neck of the bottle. In my opinion you are better off drinking all the wine (or maybe using it in your cooking).

It's not the cork that causes the wine to oxidize. It's the oxygen in the air that does that. You have to remember that the wine is in constant contact with the cork while the bottle is stored, and in fact that contact is what prevents the wine from oxidizing.  For more, check this Q&A thread.

The problem, from my experience,  is that cork plugs that fall into the wine are usually somewhat broken, and small pieces of cork are disconnected from the plug and float in the wine. So, what you need to do is to pour the wine, as lilly suggested to a carafe, but to do so through some sort of filter that will prevent the small cork debris from moving to the carafe to.

Take a square cloth napkin and fold in a triangle lapping pointed part of the napkin.  Slowly put the pointed part of the napkin into the bottle trying to unfold the pointed end of the napkin.  Shake cork into pointed end of napkin and slowly pull it out of the bottle.  This is used if you want to save your bottle for future use.