New gaming systems seem to come out every couple years. This presents the problem of what to do with old games and systems. There are several things you can do with both working and broken video games and consoles.
If your console is still working, consider selling it. Ebay is a great place to sell old consoles. Just be sure to package it safely when shipping. Craig's List is another great place to do this. Additionally, many video game stores will accept working systems for cash or store credit towards new games. There is actually a pretty good market for old systems.
If the console is broken there are still ways to make money. Two of the three video game stores in my city actually pay you for a broken system. I got $20 for a broken Nintendo DS that I found at a park and I got $40 for my old XBOX 360 when it broke and was out of warrany. My guess is that these companies refurbish the systems and resell them at a profit, but I'm not exactly sure. Check with your local stores.
I've been told that for a fee Staples will recycle old video game systems and games, but I can't personally verify this.