It sounds like you've found a ground conductor. This would be a single conductor, usually a stranded wire made of copper or aluminum, clamped to the pipe. May or may not be in conduit. If you are able to follow the wire you may find it is used to bridge an electrical discontinuity in the plumbing (you'd see this across your main water meter) or to carry a ground connection to another piece of equipment (such as a telephone termination box).
For providing ground to your main service panel, the panel is usually connected to the water supply line if that pipe is metal. This is typically done inside the house, particularly on older houses. If your water supply line is plastic, you will have grounding via other means such as ground spikes driven down either outside or through your basement floor.
There's no likely danger from getting a ground conductor a little wet once or twice. If you've located the leak on a pipe and are just really narrowing down the exact location, leave the electricity on as long as you don't expect water to reach any electrical parts that carry a current - switches, outlets, lights, etc. You can touch a ground conductor, wet or dry, safely as long as you're not also touching a current-carrying conductor. You should NEVER be touching a live current-carrying conductor because you're often touching a ground (light housings, bathtubs, flooring), because current seeks ground, and because if the path to ground leads a lot of current through your body you will have a very bad day.
If I had a rough idea where the leak was, and didn't think there was serious risk of water leaking into or spraying a live electrical part, I'd probably leave the electricity on. Lower chance of banging into or tripping over something, which is good.
If you're not sure, get a non-contact voltage tester and you can check if there's live wiring near the pipe.