Q:

What is Bitcoin?

I'm a programmer, and a generally smart person, but I can't understand really what bitcoin is. 

Is it a new payment merchant like PayPal?
Is it a virtual currency like LindenDollars that can't be really used as money? (put money In, but not Out)
Is it a new currency system, where people exchange bitcoin addresses instead of real money?

A:

Is it a new payment merchant like PayPal?

No.
PayPal is a company, Bitcoin isn't.
PayPal deals with existing currencies like US dollars, Bitcoin doesn't.
PayPal holds your balance for you, can freeze your account and can stop you from sending your money to people it doesn't like.  With Bitcoin, nobody can freeze your account or tell you how to spend your bitcoins.

Is it a virtual currency like LindenDollars that can't be really used as money? (put money In, but not Out)

Whether it is a currency or not isn't clear, but it is virtual.  Some say it's more like a commodity than a currency.  It acts a lot like gold, in that there is a finite supply of it, it's valued by some people, and it can be used as money anywhere that accepts it.  Also, it can be bought and sold for US dollars and many other currencies.  At the time of writing 1 bitcoin is worth 470 USD, and there are plenty of people willing to buy them from you at just under that price or sell them to you at just over that price.  The price is decided by the market. The price bitcoin trades at is the one people think it is currently worth.

Is it a new currency system, where people exchange bitcoin addresses instead of real money?

It acts a lot like a currency and can be used to trade with anyone who accepts it.  It is very cheap to transfer between any two points in the world, and very fast compared to transferring US dollars or other currencies.
See this introductory video for a slicker introduction.

A:

It's a distributed digital cryptocurrency. The key word here is distributed, i.e. there is no central authority, instead transactions are handled peer-to-peer with a complex protocol where everyone ensures no one cheats.
New bitcoins are created at a predetermined rate to people who run "miners", which are a vital part of the protocol that checks for transaction genuinity.
You can exchange bitcoins with other hard currencies using public exchanges such as Mt. Gox. You exchange them with other people willing to sell or buy them, just like on a stock exchange.
For more details: http://www.weusecoins.com

