The Nook Color is not a general purpose Android Tablet. It is an eReader. You cannot add .apk files like you can on Android Phones, and some other tablets.
B&N is planning on allowing some new applications to be added via a Marketplace on the B&N website starting in several months. However, the number of appications and the types of applications will probably be somewhat limited.
You cannot install your own applications to the default Nook Color. There are only two options.
1. Root your Nook Color. There are various posts on the Internet that show you how to do this. This will allow you to access the underlying Android system, which means you can install whatever you like. HOWEVER, doing so will most likely void your warranty for the device. So you are doing this at your own risk. In addition, any software updates provided by B&N in future may not be installable on the device or may render the device useless (hopefuly not). So, if you want to go down this path, make sure you understand the risks.
2. Wait for B&N to start their own App Market. This apparently will happen in the next few months. However, as the other post mentioned, the options may be limited (although I would think that a lot of the free apps will be available)
Personally I don't agree with the model of limiting the underlying hardware capabilities due to a software layer, but that seems to be how the tablet industry is shaping up. It's a shame, really.