What Is Proof by Casework, or Exhaustion?

In the snail park, there are two sets of four parallel paths, where the lines that are not parallel intersect each other at right angles. You want to place a snail at one endpoint of each path, so that he crawls to the opposite endpoint of the same path. If all of your snails crawl at the same speed, how many different ways can you place them, so that no two snails ever collide?