There are different reasons why your hot water heater can leak, and we will review some of them below.
One cause could be the temperature relief valve. The purpose of this valve is to relieve pressure from the tank in the event the pressure in the tank has built up too high. This valve is located at the top of your water heater. If the water temperature of the tank is set too high, or the water pressure itself coming into the tank is too high, then the pressure within the tank itself becomes too high and this valve should open up and let water out. If this is where the leak is coming from you need to immediately contact a licensed professional.
If there is too much pressure in the tank, in a worst-case scenario the tank could explode, which is obviously a very dangerous situation. Or it could mean the valve itself is defective and needs to be replaced. In either case, call in a professional.
If the source of your leak is coming from the drain valve (located at the bottom of the water tank) tighten it down and see if the leak persists. If it does then this valve needs to be replaced.
Are the leaks actually coming from the pipes feeding into the water heater? If so tighten down the fitting where the water leak is coming from, but carefully.
Finally, if you have a gas water heater, which most of us do, check out that the vent or flue, which releases the gas fumes, has no clogs. This is unlikely but still needs to be examined.
If the water dripping to the floor is actually come off the tank itself, it could be just condensation, which means there is no problem. You would normally see this when the weather is particularly cold, which makes the incoming water that much colder. However as the tank warms up this condensation goes away.