Negative ions have picked up an extra electron which changes their electronic charge a bit. If you've gone outside after a thunderstorm and felt like the air seemed different, it was because the air was filled with negative ions.
Looking at negative ions by themselves you're missing the big picture, there are so many items in our lives that pump out positive ions, particles that have lost an electron, TVs, computers, cel phones, game systems, etc. There's even a new syndrome that is being tossed around - Video Operator Distress. Apparently too many positive ions in your environment will weaken and make you sick.
As with so many things in our lives, balance must be achieved. If you have a lot of positive ion producing items around you then get rid of them or balance them out with something that produces negative ions.
Will this help depression? Depends upon the source of the depression. If it's physical then perhaps the negative ions will help. I know I always feel better when I go out after a thunderstorm to feel the life in the air.