Corners
Occupying a corner square gives you a disc that can’t be ever be captured by your opponent. You can use this as a foundation to extend outwards to build a block of discs safe from capture.
In contrast, the square diagonally adjacent to a corner is usually a dangerous square to play on and is best avoided. This is because it is hard to defend, and your opponent may be able to use this to capture the corner.
Mobility
Mobility is a measure of how many moves are available to you. Low mobility means few choices, and high mobility means many choices. It is important to try to keep your mobility high, especially in the early phase of the game. This gives you the freedom to play where you want, and helps to avoid being forced into making bad moves.
Ultimately Reversi is won by having more discs than your opponent. However, early in the game mobility is often increased by having fewer discs than your opponent, and with those discs inside a block of the opponent’s pieces.
To see this in action, start a game with the computer playing against itself with one set to Expert, and the other set to Novice, and make sure Highlight Legal Moves is turned on. You should see the Expert player with fewer discs than the other player during the early part of the game, and it should also have far more legal moves available.