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Turn off the water at the valves beneath the sink. If there aren't any valves beneath the sink, shut off the main water supply. It's also a good idea to plug the drain or close the pop-up so small parts won't fall in.
Remove the handles from the faucet. This may involve loosening a setscrew or removing the cover plate to access the handle screw.
Next, loosen and remove the bonnet nut holding the faucet stem in place (Image 1).
The faucet stem has a raised ledge on one side. Make a note of the direction the ledge faces and remove the faucet stem (Image 2).
Fish out the packing and spring assembly with needle-nose pliers.
Install a new spring, small-side up, and push the new packing into place on top of the spring. This part can sometimes be a bit tricky, so be patient.
Replace the stem and reattach the bonnet nut. To reverse the direction the handles turn, reverse the direction of the ledge on the stem.
Reattach the original handles, or install new handles to give the bathroom a different look.