For reference, what "stealth mode" does is very specific--it means that if something requests a connection on a closed port, rather than responding "Sorry, that port is closed." the computer won't say anything at all.
Which, on the other end, appears as if there is nothing there. So far as I understand, it's basically a deterrent for some kinds of network attack, because an attacker that isn't persistent (or one that is, if you have no ports at all open) won't even be able to tell if they're prodding at a computer, or if there's nothing at that address.
As already said, if you're already behind a home router, it won't break anything, but it also doesn't really do anything useful--outside access is already blocked at the router apart from any ports you've forwarded. I'm not sure whether it interferes with auto-discovery or not--Bonjour might open a hole even with it on--but it's a mostly-unnecessary level of paranoia. By the time someone has gotten past your router, stealth mode isn't going to provide much extra protection from what they could try anyway.
Now, if you're getting a "raw" connection directly to the internet or other public network--for example, a university wireless network--it's probably a good idea, because you can never be too paranoid when exposed to a large, unfriendly network.