There are about 5 steps to this and it's easier if you have a breakdown diagram.
The pictured hose bib is a Prier, probably a C-138
Two for the price of one, first section is how to replace the seat washer, second to repair stem leakage.
Note: The Handle, Stem, Stuffing Box, Packing Nut, Packing Nut Seal and Seat Washer all come out as one complete assembly on this faucet.
Replacing the Seat Washer, Faucet leaks when fully closed.
Find the water shutoff for the zone this faucet is in (mains for whole house if that's the only one) and shut the water supply off.
Turn the hose bib handle 1/4 turn open.
Loosen the Packing Nut (see illus below). As you turn it, the Stem will probably turn with it, if not, turn both Stem and Packing Nut in the same direction till the Stem/Packing Nut assembly comes out of the faucet body.
Remove the Seat Sealing Washer screw, and replace the washer.
Reverse operation for assembly.
If the faucet is leaking around the handle, gently tighten the Stuffing Box Nut 1/16-1/8 turn to compress the packing (what the stem seal is called). If it feels like you're tightening metal to metal, the packing has worn out and needs to be replaced.
Replacing the Packing, Faucet leaks around the stem when open.
Remove Handle Screw and Handle.
Remove Stuffing Box Nut (also called a Gland Nut).
Remove Packing and replace with new packing washer or graphited packing cord.
Replace Stuffing Box Nut (finger tight)
Replace Handle and Handle Screw.
Make sure you have a hose and closed nozzle attached to the hose bib.
Open faucet to pressurize hose and tighten Stuffing Box Nut down gently until the stem stops leaking.
Try tightening the packing nut (the hex part the stem goes through) a little. That's what it's for. Don't overdo it. The "packing" is the material that seals around the stem - the packing nut compresses that material. On a valve that is not used often, actuating the valve does commonly cause it to leak, as things have "set." Tighten a little, wait, tighten a little more, wait - try to sneak up on it, as the pressure is a balance between enough to seal and little enough that the valve turns easily.
Best Answer
The term leaking is somewhat ambiguous, in that the hose bibb can leak at the stem or out the spout. Leaking at the stem is normally due to a worn or cracked washer, unless its quite old and that may require using "stem packing", namely a waxed cord material wrapped around the stem instead of a more modern washer. If the leak is more of a drip out of the spout it could be caused by a bad seat or worn stem that either will wear when over tightened.
Some bibs have a seat, normally brass, that can be removed with an Allen wrench, but most don't and may require the stem to be replaced. Most stems are pressed in past a bulge point that acts both as a safety feature, so that if you were working on the bib with the washer retainer nut off and loosened the stem where the threads were no longer engaged, the stem won't fly out under 100+ psi water pressure. So, if you fouled the seating area of the stem in most cases you can't get the stem out, your only real option at that point is spend the under $5 and replace the bib with a new one.