Instead of just saying "follow the instructions in your kit" in an article on fixing flats, let's review a little more in depth with the most common type of patch kit, which uses vulcanizing glue (all the vulcanizing kits are pretty much the same, though I've heard good things about the Rema Tip Top TT-02).
It should include patches, glue, and sandpaper at the least. I also store spare valve caps and a presta-to-schrader valve adapter in the little box. The instructions often have just pictograms (if there even are any), but there are important nuances that shouldn't be glossed over!
- Once the leak in the tube has been found, rough up the area with sandpaper (to expose fresh rubber) and blow away dust, doing your best to keep things clean
- Apply a small amount of glue to the tube (not the patch or tire), spread a thin, even layer with a clean-ish finger, and then wait 5 minutes—I think the problem is that no one does this
- Now the area will no longer be tacky. Remove the foil backing, apply the patch, press firmly, and wait another few minutes before inflating. Don't put any glue on top of the patch or in the tire.
For extreme detail on preventing flats and finding root causes, look no further than Sheldon Brown's site.