The length of a crop is designed to allow enough leverage for it to be accelerated rapidly with a controlled flick of the wrist without causing the rider balancing problems. Thus, a true crop is relatively short.
The term "whip" is a more common term that includes both riding crops as well as longer types of horse whips used for both riding and ground work. A whip is a little slower than a crop, mostly due to having slightly greater length and flexibility.
Crops are designed to back up the natural aids (leg, seat and voice) of a rider.
While it is considered proof of bad horsemanship and abusive to whip a horse painfully, the crop's termination at a small leather loop indicates a design intended to be less painful than a device causing lashing, such as a cat o' nine tails. Even given this moderation, the crop is properly to be used very rarely.
The riding crop is still widely seen as the traditional symbol of dominance in BDSM activities. Its material is the only obvious link with the (stereotypical) leather fetish in that scene.