The gameboard is a grid of buttons. You click on a button to signify that you know there are no mines on that spot. When you click a button in the grid, one of three things will happen:
1) You have clicked a blank mine. When this happens, each of the surrounding eight buttons are also 'clicked', because there are no mines in the 8 surrounding squares. If they are blank also, this process continues in a chain reaction until there are no more blank mines.
2) You have clicked on a button adjacent to a mine, in which case the number of adjacent mines will be displayed as a number in that grid square. Just think of blank mines as the number 0. Adjacent mines are mines that are in the eight surrounding squares.
3) You have clicked on a mine. Game over, please try again.
Using the process of elimination and such, you can usually decipher where the mines are, but not always.
The point is to mark all the squares that contain mines (by clicking the right mouse button to place a marker). Once you currectly mark all the mines in the grid, you win.