If you can feel the upper bearing of the headset shifting during the above test, the bearing needs to be adjusted. Some headsets, such as older ones by Mavic, have a lock screw that needs to be loosened first, but normally it's a matter of tightening the nut that contains the upper bearing race against its lock nut. For this you need two appropriate wrenches; don't try to do this with regular household adjustable wrenches!
Unscrew the locknut, tighten the main nut until there is some resistance when turning the handlebars, then tighten the locknut. Tightening the locknut will loosen the main nut slightly, which is why you need to tighten it a bit too much at first. I usually have to repeat this a few times until I am completely satisfied. The handlebars should turn without resistance but pass the finger test, see above.
When I have to adjust my headset, I always use the opportunity to clean and re-grease all parts, both in the upper and lower bearings. A lot of dirt accumulates near the lower bearing, and some might find its way even into a properly sealed bearing.
Sometimes the result is the "indexed" mode where the handlebars seem to snap into the forward position. This is an indication that you waited too long. If you have plain bearing balls, there is one last chance to fix:
Bearing balls are either loose or in a cage (a small ring that holds all bearing balls). Since the indexing effect happens because all the balls simultaneously fall into the pits they have pounded into the races, it helps to change the number of balls. The best approach is to remove the cage if you had one, or put one in if you didn't. If you remove the cage, you must add more bearing balls - add balls until the bottom of the race is completely filled (liberally grease the race to hold the balls in place), then take one back out.
If you didn't have a cage and can't easily find one, it sometimes helps to remove a ball or two and hope that the balls float around enough to avoid alignment with the pits, but never fix a caged bearing by removing the cage without adding more balls. There simply won't be enough of them. Don't add fresh balls, replace all of them because the old balls will have been slightly flattened over time while the new ones are still round. Either way, these fixes are temporary and you should consider a new headset as soon as possible.
See also technical articles on "The Drive Train"
