Skill
Depending on the game, the player may need physical skills, mental skills or social skills -- which may be further sub-categorised (e.g. mental = memory, spacial, pattern-matching, etc.). When designing a game, Jesse Schell recommends making a list of the specific skills that will be required.
This is more important than it sounds. For example, Big Brain Academy for the Wii (one of the major "brain training" games popular five years ago) covers a range of mental skills; however it also requires users to accurately point a Wii remote at the screen to control a cursor, which must be moved at speed to select items in a timed game. These two skills are very different, and people who are good with their minds may struggle with the physical aspect and become frustrated. It's apparent that the makers of the game felt it was a risk worth taking to appeal to a more mainstream audience.
Chance
Schell argues that games need some element of chance to be fun. He links skill and chance, explaining how skill levels can introduce an element of chance (otherwise games like ten-pin bowling would be pretty boring).
His most important observation is that expected value is important to the player. This is the average 'win' for the player. In most casino games, the odds are stacked to ensure that a player will, on average, walk away with less than they stake -- not much of an incentive!The skill of a designer lies in making this more attractive to a player, by balancing the odds carefully. (In the case of a casino, this average loss is kept to an optimal level to avoid deterring players; they also apply clever psychology to emphasise high potential winnings as a counterbalance.)
Expected value can be used to motivate a player: if they feel they should have won, yet have lost, they feel a need to prove themself. In the case of Windows solitaire, the 'Vegas' scoring system is carefully balanced to encourage a player to play another round in an effort to recoup their losses.
What have we learned from this? Well, skill and chance play an important role and cannot be ignored. Most importantly, a poor balance of skill or chance can scupper a game.