Re: leaking kitchen faucet
Single handled faucets generally use a cartridge. Depending on the brand and style faucet, you either replace the seals (cups and springs ) or the entire cartridge.
You will need to determine the brand of your faucet. From there you either do some research to figure out which model it is to purchase the right parts or disassemble the faucet and take the parts to the store to match up replacements with.
Don't forget to turn off the water at the valves under the sink before you start the repair.
e: leaking kitchen faucet
Spruce is correct; there are numerous sites on the web that illustrate how to repair a single handle kitchen faucet.
Google "kitchen faucet repair" or "single handle faucet repair"(followed by the name of your faucet, if you have it; Delta, Peerless, Moan, etc); Google the phrases with and without the quotation marks.
Some sites are below; conduct your own search until you find an exploded view of a faucet that looks like yours.
Your should be able to find an exploded view on the internet if you can find the name of the faucet (printed or stamped somewhere on the housing).
If not, another excellent resource is the public library; the home improvement books in the 643.7 numbering section always have an exploded view of the several types of single handle kitchen faucets.
Before you start disassembling, scrunch a wad of paper toweling into the sink cavity to avoid losing any small parts.
You'll need a small hex wrench to loosen the little hex nut at the back or front base of the faucet handle so you can pull the handle to get at the faucet guts.
If this ends up taking a while, the bathroom sink faucet can be used temporarily for water & cleaning.
If the bath sink & kitchen sink are not piped seperately, consider installing separate shutoffs as a first step so you can use the bathroom sink while you make the repair.
Re: leaking kitchen faucet
Around these parts, 3 out of 4 kitchen faucets are single handled Deltas. Very popular as regards price and performance. Easy to rebuild also.
Yours might also be a Delta. If you remove the handle and see that it looks identical to the one pictured in the linked site, then it likely is. (Peerless SH faucets will look identical and most of the internal parts are interchangeable with Delta......although you'll likely see rebuild kits labeled for Peerless.) You can find the rebuild kits for these Deltas at most any hardware store or big-box.
If yours drips from the spout after being shut off, then the odds are that one of the little rubber cup seals needs to be replaced. Replace both off them and their springs.
You may find several different kits available for these faucets, each containing a different number of parts. The price won't be significantly higher and so I'd suggest you buy the complete set and replace everything in there while you're at it....including the large o-rings. (See sidebar to article) Buy a little container of plumber's silicone grease also and lube all the parts a bit as you install them.
Be especially careful when you reinstall the ball so that you get it oriented properly. Pay attention to the slot on the ball and the locating pin for it inside the faucet body.
The kit will come with a little forked wrench so that you don't have to use a screwdriver like the guy in the linked article did.
If yours doesn't look like the pictured faucet when you remove the handle......you've got a different critter/brand/type.