There's a simple strategy at Reversi that will work wonders against the average player

Reversi Strategy

Pieces are flipped very quickly and easily, so it's not very important, and in fact typically detrimental to try to gain a majority of pieces early in the reversi game. Corners, mobility, edge play, parity, endgame play and look ahead are the key elements of good Reversi strategy.

Corners, once played, cannot be flipped, and can be used to anchor groups of pieces (starting with the adjacent edges) that also cannot be flipped. So capturing a corner is often an effective reversi strategy when the opportunity arises.

An opponent playing reversi with reasonable reversi strategy will not so easily relinquish the corner or any other good moves. So to achieve these good moves, you must force your opponent to play moves which relinquish those good moves. The best way to achieve that is to reduce the number of available moves of your opponent. If you reduce the number of legal moves your opponent can make, then sooner or later she will be forced to make a move undesirable to her. An ideal position to be in is to have all your pieces in the center surrounded by your opponent's pieces. In such situations you can dictate what moves your opponent can make.

When moves seem equal with respect to what moves you will leave yourself and will be open to your opponent, playing a minimum piece strategy will tend to be more beneficial. This is because minimizing your discs will tend to leave fewer discs for your opponent to flip in subsequent moves of the game. One should not play the minimum disc strategy to an extreme, however, as this also can quickly lead a lack of mobility.

While playing pieces to edges may seem sound (because they are not so easily flipped) they can often be detrimental. Edge pieces can anchor flips that affect moves to all regions of the board. Because of that, this can, sooner or later, poison later moves that you make by causing you to flip too many pieces and open up many moves for your opponent. However sometimes playing to an edge which cannot easily be responded to will leave your opponent with significantly fewer moves than any other moves.

The square immediately diagonally adjacent to the corner (called the X-square) played in the early and middle game is typically a guarantee of losing that corner. Playing to the edge squares adjacent to the corner can typically lead to tactical traps involving sacrificing one corner, or simply playing out the edge in a specific sequence.

In general you should avoid edge play in the early and middle game if you can avoid it, unless you can gain larger concessions in terms of mobility or a mass of unflippable pieces.

As reversi play progresses regions of the board will typically section themselves off, where neither side can prevent the other from playing arbitrarily into those regions. By simply counting out the number of squares in a region, one can notice if it is odd or even. If it is odd, then by playing there first, you can force your opponent to be the first to play outside of that region. This is achieved by simply playing into that region any time there is an odd number of squares available, and not playing into it when there is an even number of squares. If you take into consideration certain squares in a region that seem to be very bad (like an X-square or an edge square that leads to an obvious trap) then you can either force your opponent to play elsewhere or concede to playing one of these bad squares.

Like any good strategy for chess, or checkers, it is not sufficient to only consider the current situation on the board. For each move you consider you must consider possible responses from your opponent, then the subsequent responses you will make to those moves and so on. The aspects of the current position may not be relevant a few moves down the road. So when optimizing your mobility, gaining corners or anything else, you should consider how best to do this for the long term rather than just for the next move.

For the end game (the last 20 or so moves of the game) the reversi strategies will typically change. Special techniques such as sweeping, gaining access and the details of move order can have a large impact on the outcome of the game. Unfortunately at these late stages of the game, there are no hard set rules. The best you can do is trying to look ahead and get a feel for what will lead to the best final outcome.

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