My neighbor gave me his wireless router password to use on my smartphone. Using this, can I access any of his accounts whether it's his bank account or
Att/Direct TV account?
I ask because I think someone stole the password from me.
Probably harder to do with your phone than with a laptop, but attacks can be (and have been) mounted with a phone. Your neighbor should definitely change his password just to be safe, but IMO his chances of being attacked aren't as high as one might expect.
The attacker would have to find the location of the network if he didn't know it already, and then he'd have to get physically close enough to get onto it before the password would be of any use to him. Heading over to the nearest fast-food joint and sniffing on the free Wi-Fi there is much easier. Even cracking Wi-Fi passwords (particularly WEP, although that's not in widespread use anymore) is probably easier than actually moving around (at least for me). With cracking, you have the added bonus that anyone whose network is easily crackable is probably pretty lax about security, so additional opportunities are likely to present themselves.
To learn more about this, I highly recommend using google. There's a lot of information on the internet, or so I'm told.