After reading the thread with the title "Kitchen sink tap, washer replaceable?" I thought of writing this about my experience, today. My kitchen faucet started to go drip, drip, drip...
(It is very similar to the photo @giddyup has posted (made of metal). The inside part too looks the same, as what @johng has posted.
I dismantled the thing and saw a lot of whitish lime formation in its innards. I sprinkled some Bicarbonate of Soda and added a little vinegar - it starts foaming. Kept it for a couple of minutes, scraped lightly with a thin scraper (or a screwdriver would do), and rinsed it. Repeated this procedure twice, and reassembled - (maybe if I kept it soaked in that solution for a couple of hours, it would have been better).
So far no drips and seems perfect!
Convenient, simple and cheap procedure, anyone could try.