During a game, you can move pieces with the mouse.  First click on the piece you want to move.  The piece’s tile will shift down to show that it is highlighted.  Now click in the empty tile where you want the piece to go.  To continue moving the same piece, simply click in another empty space.

If the move was a single step, the number of moves (to the right of the board) will increment immediately.  If the move was a jump over another piece, the moves counter will not update until you end the current move by clicking off the board, clicking on the highlighted piece, clicking on another piece, or moving the highlighted piece a single step.

See also: General --> Moves.