Many gamers are motivated by the challenge a game can present. Success in a game may be governed by any of a wide variety of abilities. A First Person Shooter requires twitch reflexes, a steady hand and the ability to remain calm under pressure. A word puzzle game may require an extensive vocabulary and the ability to rethink the uses of old words, but no measure of speed. A sports simulation might well require an in-depth knowledge of the topic, in addition to arcade skill, but is unlikely to have terribly much concern for linguistic acumen.
The common thread is that all of the games challenge some subset of a player's abilities. This challenge can be a powerful motivator. The Challenge Motivated gamer is drawn to a game that tests their skills, preferably one that tests them to their limits. The gamer may also be motivated by the natural improvement that comes from working at peak. They are driven then, not only to excel, but to improve. Challenge Motivated Gamers thrive whenever a game pushes their skill set of choice, but may be disinterested in games that fall too far away from the target.
Competition is a close cousin of challenge. Many gamers are driven by the need to prove they are the best, to be pitted against their fellows and come out on top. Competition minded gamers range from those looking for a challenge in a fair fight to the sort of win-at-all cost let speaking infants that give us all a bad name. Competition can be easy to take too far. There is nothing inherently wrong with being driven by competition. To some extent, competition is merely challenge taken to the extreme. It is only when it leads to mistreating your fellow player that it begins to become less a motivation and more an unfortunate personality quirk. Competition Motivated players thrive on those games where they are pitted against one another with the outcome dictated by skill at playing the game. They will often wane in those environments that either require cooperation, such as many MMORPGs, or in games where skill plays a much smaller role, such as in less sophisticated card or dice games.