Skills are a key aspect of Hattrick. The effect of each type of skill is described in the corresponding Wikipedia entry.
In Hattrick, skill levels are shown as the terms ranging from disastrous to divine. In many situations it is convenient to use numerical skill values instead of terms, and these numbers can be found here.
Skill levels on Hattrick are always rounded down. If you are interested in exact skill levels, check the subskills page.
Stamina is absolutely vital for any players who use playmaking. If a player does not have enough stamina, a serious drop in their playmaking contribution will be noticed, particularly during the second half of matches.
Playmaking is the most important skill for all midfielders, especially the inner ones. It is also used by defensive forwards and offensive defenders. Offensive wingbacks use a tiny bit as well. As noted above, this skill is used in conjunction with stamina.
Almost every successful team needs good playmakers or else the team won't have a lot of ball possession, which will lead to fewer scoring opportunities. The only exception are teams who specialize in using the Counter Attack tactic.
Wing skill refers to the ability to create scoring opportunities by advancing down the sides. This is obviously most useful for wingers, although Wingbacks also profit from it. A forward toward wing will also benefit a lot from this skill.
The ball is supposed to go into the net. Most important ability for forwards. Any player with good scoring skill combined with good set pieces skill can score on a long shot special event.
The scorer gets the credit, but the player giving him the decisive pass is just as important. Both forwards and midfielders, including wingers, profit from being good at this. Passing is also very important for use with tactics. It helps to compute ratings for Counter Attack, Attack on Wings and Attack in the Middle.
Inner midfielders, and forwards contribute to the central attack rating by their passing ability. Wingers played offensively, normal or defensively use their passing to contribute to side attacks. There is some debate about the contribution of passing on a winger towards middle.
It is also relevant on all players when using the Pressing tactic.
One player is appointed for Set Pieces. He'll take all free kicks, penalties, and in case of a penalty shoot-out, he'll take the first penalty. In all cases, lots of skill is needed in this area.
There is also the possibility of a long shot Special Event related to set pieces.