Patio stones sink from unstable (soft) ground, normally caused by improper drainage. The only way to permanently correct this is to stabilize the ground and allow the water to drain away. If your patio is at the bottom of a hill or in a depression in your yard, you might want to think about re-landscaping your yard.

[edit] Steps

Remove the problem stone and all stones next to it.
Dig down into the ground about 3" (If you're not that strong, rent a "Bobcat" (a small excavator), to do the job.
Backfill with 2" of coarse stone to allow proper drainage. Even out as level or close to level as possible.
Backfill with 1.5" - 2" of coarse sand or fine gravel
With a 2x4 or other straight board scrape the top layer smooth using a level to guide you. Let the sand sit slightly higher then the base of the other patio stones to allow for settling.
Place the patio stones back in their respective places.
Use some sand and a broom to fill in the gaps between the stones to keep them from shifting when in place, this also helps by reducing the weeds from growing up between the stones.

[edit] Tips

The stones will settle a bit after this repair, if the stones go lower then the desired final resting place:
1. Remove the stones
2. Add more sand or gravel
3. Smooth out with a level
4. Replace Stone
Fine gravel works better then sand as a finishing layer as it does not wash away with water as easily
Also Works for bricks or tiles. Make sure you cover a wide area. If you use brick, remember that if the area is shady, moss will grow quickly, causing slippery conditions and they'll have to be cleaned often.

[edit] Warnings

This requires a lot of time, and can be more labour intensive than expected.
NOTE: Before digging, even if you think it's safe, contact your local power and gas companies (for example, in one region it's 1-800-DIG-SAFE), to make sure you won't hit any electric or gas lines, which could be deadly! Also check your lot plan to see where sewer lines run. Cutting into sewer lines is messy, a bio-hazard and very expensive to fix!