In American football, the only players who cannot catch a pass are the interior offensive linemen. They are considered "ineligible." Tight ends, split ends and everyone behind the line of scrimmage (including the quarter back, punter and kicker) are eligible receivers. Ineligible receivers may not cross the line of scrimmage on a pass play until the ball is thrown. They also may not be the first player to touch a pass, but they may catch a pass that has already touched another player.
Of course, any defensive player can intercept a pass at any time.
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The running backs and wide receivers can catch the ball and so can the quarter back if it is a halfback or receiver option designed to go to him. The tight ends can also catch a pass too. A lineman can also catch the ball but that is only if it touches a defender first.
To be more specific: By rule, the eligible receivers are the two players on either end of the offensive line (usually wide receivers or tight ends) and any of the players in the backfield (including all backs and flankers who are at least one yard off the line of scrimmage).
Any one on the field can catch the ball really. It's primarily the wide receiver or the tight end.