Why does my front mountain bike wheel sway left and right slightly?
Two reasons why you may be seeing the wheel move from side to side, and each has a fix that involves tightening something.

The first is that the rim is out of true, so when you spin the wheel the rim and tire appear to move side-to-side relative to a fixed point such as the fork or rim brakes. What your mechanic would have done in this case is adjusted the tension in left and right hand spokes to make the rim true again. YOu may have had a single spoke that came loose and needed re-tensioning. If you are still seeing some apparent movement when you spin the wheel, or you have rim brakes and the rim hits the pad at some point the mechanic may simply have not done a particularly good job. You can ask for them to improve the wheel or go to a different mechanic.

The other possibility is that the hub bearings are loose and the whole wheel is wobbling on the axle. This is easy to diagnose: grab the tire and try to move the wheel side-to-side, if you can feel the wheel rocking slightly the bearings are possibly loose. Some hubs have adjustable bearings and you mechanic may have tightened them. Again you can return to your mechanic or use a different one.