If there is a visible leak in the faucet above the bathtub, then the fault probably lies in the “O” ring or washer inside the faucet. You should first turn off the valve that supplies water to the bathtub faucet. Then, you can simply unscrew the end of the faucet from where the water spouts out, and remove the washer.
You can find a new washer in your neighborhood-plumbing store. In case the washer has been badly mutilated then click a photo of your faucet and show it to the sales person. Chances are that he/she might recognize the brand of the faucet and hand you the correct washer. Install the new washer and screw the faucet back on. If you find any solid gunk that formed a clog in your faucet then remove it before screwing the faucet back.
The tough part is where there is no visible leak but you still find water dripping from below the bathtub. If your bathtub is without any liners or ceiling, then the leak should be easy to locate and repair. If the bathtub drain is leaking, then wipe the area around the drain and fill up the bathtub with water. Very soon, you might be able to locate the point from where water is dripping out.
In case your bathtub is enclosed by liners or has been mounted on a ceiling, i.e. the small space between the floor and the bathtub, then things could get really ugly. You will need to rip out the ceiling drywall and try to locate the leak.
If you have no experience in plumbing then it would be wise to contact your nearest plumbing contractor. Get an estimate of the cost, material and time involved. If water has seeped into the ceiling then there are chances of mold developing in those moist areas. Just ripping out the drywall and re-caulking might leave the mold to dangerously spread its tentacles. You will need to remove everything and preferably use a hot air gun to remove the moisture from underneath the bathtub before sealing it back.
If the pipes below the bathtub are leaking due to loose joints then you will need a plumber’s wrench to tighten it. If the leakage persists then you will need to use teflon tape on the threads to stop the leak. If the outlet pipe is clogged then you can try using chemical cleaners to unclog it. For major leakages under the bathtub, a plumbing contractor would be the best since the leakage might require repainting the ceiling on the floor below the bathtub too.
If you ignore the small leak in your bathtub with the hope that it will stop, then you might just end up with a hefty repair bill in your hand. Try to locate the fault and fix it up immediately, or hire a licensed plumber or contractor to come in immediately to fix it. A small leakage can quickly result in mold infestation if you do not act on time.