If your outdoor faucet leaks around the handle when you turn it on, you either have a loose packing nut or a bad packing washer. First try tightening the nut with a wrench or pliers. (The size and type of nut varies a bit with faucet styles.) If the nut is tight but the leak persists, shut off the water to the faucet inside the house, remove the faucet handle from the outside faucet and unscrew the nut. You should be able to pull off the old packing washer and take it to a hardware store to find an exact replacement.
Older faucets may have a wad of string, called valve packing, instead of a packing washer. If so, pick up new packing (graphite-coated string) at the hardware store, unwind the old packing and wind on the new clockwise. The packing nut should compress it tightly. You may have to wind on one layer, tighten the nut and then repeat the process to fill the space around the stem completely with packing string.
Better Pipe Cutting
It's difficult to cut a thin metal pipe, such as a P-trap, with a hacksaw without squashing the pipe or mangling the cut. To make a nice cut, insert a section of wood closet rod or handrail into the end of the pipe to be cut. Wrap one end of the rod with tape to fill out the rod to the pipe's inner size. Clamp the wood in a vise and cut through both the pipe and the wood.