Some of my favorites:
Orange - It's about a group of people losing their friend to suicide, after his own mother did the same. They send letters back to their past selves with advice on how to prevent this.
Welcome To The NHK - About a shut-in who deals with depression, paranoia and delusions. His best friend also deals with it.
Paranoia Agent - This is one of my favorite, favorite anime's ever, it messes with your head a lot. This plot isn't centered around depression, but one episode focuses on a group of people who make a suicide pact over the internet.
Sorta played for laughs since one of the members turns out to be a 12 year old, and they keep trying to ditch her while also coming up with ridiculous ways to kill themselves lol