The Riding Crop is a tool added by Forbidden Magic. When riding a mount, holding the Crop and right-clicking will temporarily buff the mount with Speed at the expense of inflicting 2 () points of damage on it. Using the Riding Crop or slaying mobs with it in The Nether will occasionally cause Lust Shards to drop.
The riding crop is, as it name suggests, a tool used to motivate mounts to move faster. Right clicking with it in your hand while riding a horse or a other mount will give the mount a quick lash of pain and cause it to move faster for a little while.
You could hit other creatures with the crop too, but what would possess you to do something like that?