First thing's first - you can't rule out that there may be live wires exposed inside the wall, so there is a potential hazard here. If you touch the live wire poking around in there, you could receive a shock, and that could injure or even kill you. It is also possible that the exposed wire could start a fire.
To be safe you'd want to turn off the breaker for the broken wire's circuit, but you don't know that it was power wiring, much less what circuit it's on. So the safest thing to do would be to turn off power to the whole house while investigating.
That's an awful lot of trouble for what may very well turn out to be nothing hazardous, but can you really take a chance with this kind of thing? I don't think so.
There are only a few likely possibilities for copper wire in your wall. It's either electrical power wiring or some low voltage wiring - phone, network, thermostat, doorbell, etc. Hopefully it turns out to be low voltage because the repair will likely be much easier.
The next step after turning off the main power just-in-case would be to look inside the wall and see if you can figure out exactly what happened and what you hit. At this point - with power off, no heat / air conditioning, no refrigerator, no lights, etc. - it may be best to just call in a pro for an emergency repair so that it's fixed quickly and safely. Keep in mind if you do find damage, you may wind up calling a pro to fix it anyway, so if you try to DIY this it may just be "prolonging the agony."
If you want to try to investigate, you may be able to see what happened with an inspection camera that goes into a bore hole. Of course you probably don't have one of these lying around the house. Even if you do, it can be hard to figure out what you're looking at in there, especially for an untrained eye.
You could open up a bigger hole to get a better look in there. You might want to cut out a hole and patch it up afterwards, or you could cut a hole for an old-work low voltage mounting ring - this way you can cover it with a blank cover plate once you see what's going on in there. If the hole you drilled is at receptacle level, you can match the height of the existing receptacles and this won't look out of place after the fact.