The center fielder covers the middle portion of the outfield (when viewing the field from home plate).
On defense: Range is imperative for a center fielder, as he has more ground to cover than either of the corner outfielders. Center fielders ideally will also possess strong throwing arms, as they often are required to throw to home, third and second in an effort to prevent baserunners from advancing.
The center fielder is considered the "captain" of the outfield and typically will take precedence over a corner outfielder if the two are attempting to field the same ball. Center fielders will often call off corner outfielders by yelling "I got it" in such situations.
On offense: With the advent of advanced stats that can quantify fielding skills such as range, clubs are increasingly giving consideration to defense-first players when filling the center-field spot in their lineups. Because of the sheer amount of space center fielders are expected to cover defensively, the position isn't typically occupied by players who have little else to offer on offense besides power. Center fielders that do hit for power tend to be well-rounded players who also possess the speed and the arm to field the position.