A subsidy password is a code that is used to unlock a cell phone that is already locked onto one particular network. This is why you have to go through a few extra steps to unlock a cell phone if you wish to use it with another carrier.
The term comes from the word subsidized. The principle is actually pretty simple and basic economics. The cell phone providers give the buyer a fairly good deal on the phone when they buy it alongside a contract for their network. They lose potential profit on the phone itself for the long term profit of the cell phone contract. In order to keep the phone restricted, a subsidy code is used.
To remove the subsidy code, you will need to use a subsidy password to unlock the phone for use with another network.
Most people learn about subsidy passwords when they try to put their old SIM card into a new phone. A number of providers automatically pull up a message requesting the subsidy password (and naturally a number of people run off to Google "what is a subsidy password?").
I don't really wish to get into the fairly complicated process of actually unlocking a phone. The worst part is that there isn't one real technique. It's usually a little different for each phone. If you're a legitimate customer of the company and can confirm that the phone is in your possession, then you should be able to call customer support and have the cell phone unlocked. Apparently some people have had luck with calling tech support and then telling them that they plan to leave the country and need the phone unlocked. Sometimes just telling the truth works.
If all else fails, there are plenty of articles on Brighthub that tell you how to unlock a cell phone. Note that we have many guides available for specific brands of phones. The one I linked is pretty good for understanding the concept of unlocking a phone though and should give you a good idea of how to get a good start.