Along the route, there are checkpoints for hill climbs (when there are hills) and intermediate sprints. Whoever passes that point first wins the points for that checkpoint. That's why the intermediate "sprint" points are called sprints - those in contention line up and at the last minute, they all sprint for that mark. First 3, I believe, get points for crossing that line first. In theory, there could be dozens of sprint points, but I think there's only 2 per race; one intermediate and the finish line. For those who know more (or have read more on the TdF site) correct me if I missed something.

Same deal holds for the hill climber (polka dot jersey); first one over the line (at the summit) gets the most points. Then 2nd, 3rd etc. I think - not sure here - there are 5 point winners in hill climbs.