I keep reading about the importance of controlling the center in an opening, however (and forgive me my ignorance), I'm unsure if control means to occupy the center squares or to position pieces such that they are attacking the center squares. If anyone could elucidate the concept of controlling the center, I'd be very grateful!
It means both. The 'center' is most common defined as d4, d5, e4, and e5. (The squares immediately around these are also sometimes included.) Pieces in the center have greater mobility and flexibility than those on the edges. The knight in the corner versus a knight in the center is the classic example. A bishop in the center may attack b8 and h8. The same bishop on g2 is still powerful, but can't immediately control the opponent's king-side.
Putting pawns in the center cramps the opponent and removes his options. If you control the center squares, you can begin thinking about working pieces into strong-points on the opponent's side of the board. Pawns on the the 5th rank can protect pieces on the 6th. These pieces are seriously in the opponent's business.
It's hard to keep the queen or a rook in the center (during the middle game) since the opponent's minor pieces will probably be able to drive them away. Also, controlling the center doesn't mean your just plop your pieces and pawns in the center, but that you can keep them there. Finally, controlling the center is only a means to an end. Once your pieces are qualitatively better than your opponent's, you must strive for the win.
Controlling the center means your pieces are placed on the board in such a way that the center 4 squares can be attacked and controlled (as shown in the picture below). When pieces are placed in the center they have more mobility and control over major areas of the board (especially with knight) than when they are placed elsewhere.
Controlling the center does not mean that your pieces have to be in the center. Fianchetto is a good example on strategic placement of bishop so that center can be covered from far end of the board.