The purpose of the game is to open all the cells of the board which do not contain a bomb. You lose if you set off a bomb cell.
Every non-bomb cell you open will tell you the total number of bombs in the eight neighboring cells. Once you are sure that a cell contains a bomb, you can right-click to put a flag it on it as a reminder. Once you have flagged all the bombs around an open cell, you can quickly open the remaining non-bomb cells by shift-clicking on the cell.
To start a new game (abandoning the current one), just click on the yellow face button.
Happy mine hunting!