Lightning, hail, strong winds, even tornado warnings, are all covered in the Rules of Golf. Here are some key rules that address severe weather conditions.
Although bad weather on its own isn't reason enough to discontinue play, you may stop without anyone's permission if you believe there's danger from lightning, hail, etc. -- but you must report having done so. (Rule 6-8a)
You can briefly take shelter, while waiting to play your next shot, but don't let someone protect you from the elements as you make a stroke. This includes having someone hold an umbrella over your head or shield you from the wind. You may putt one-handed while holding an umbrella, though. (Decisions 6-8a/2, 14-2/2; Rule 14-2a)
If wind moves your ball after you've addressed it (defined as taking your stance and grounding your club, except in a hazard), take a one-stroke penalty and replace the ball. If wind moves your ball before you address it, play the ball from its new position. No penalty. (Rule 18-2b; Decision 18-1/12)
If you mark your ball's position when play is suspended only to find that wind or rain moved the marker, simply replace it. If you can't determine the exact spot that the marker was moved from, make an estimate and replace the ball. (Decision 20-1/10)
If either your stance, swing or ball is interfered with by casual water from heavy rains and your ball isn't in a water hazard, you get free relief. Take a drop within one club-length of the nearest point of relief that's not closer to the hole. If the ball was in a bunker, that spot must be in the bunker, otherwise it can't be in a hazard or on the putting green. (Rule 25-1)
If casual water interferes with your line of putt, stance, stroke, or ball on the green, place -- don't drop -- the ball at the nearest point of relief (without penalty) that's not in a hazard. This spot may be off the green. Note: You're not allowed to remove the water. (Rules 25-1, 13-2)