Controlling the corners is vital in a game of Reversi, but it is also obvious. Strive to be the first player on the outermost ring. With your token on an edge, it is much simpler to work your way toward a corner. It is also easier to keep your opponent off of that edge than for your opponent to take it from you.
In your attempts to capture border spaces, do not rush. If possible, do not place any of your tokens on the second-outermost ring until absolutely necessary. Doing so opens up the outer ring for your opponent. When your opponent places a token on the second-outermost ring, capture it by placing a token of your own in the outermost ring if you are able. You now have a much more powerful position, and you're much more prone to take the corner because you beat your opponent to the border. The second-outermost ring is is composed of the spaces marked with a %% below.
+--+--+--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+--+--+
| |%%|%%|%%|%%|%%|%%| |
+--+--+--+--+--+--+--+--+
| |%%| | | | |%%| |
+--+--+--+--+--+--+--+--+
| |%%| | | | |%%| |
+--+--+--+--+--+--+--+--+
| |%%| | | | |%%| |
+--+--+--+--+--+--+--+--+
| |%%| | | | |%%| |
+--+--+--+--+--+--+--+--+
| |%%|%%|%%|%%|%%|%%| |
+--+--+--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+--+--+
Reversi is just as much about controlling your opponent's tokens as it is about controlling your own. The power of a Reversi space is directly proportional to its distance from the center of the board. Keep your opponent out of the outermost ring, and you are almost guaranteed a victory.
Thanks to maxClimb for pointing out that the power of a Reversi space is actually directly proportional to its distance from the center of the board. I knew what I meant and typed the exact opposite anyway.
/me can't believe he did that.