Well, the Ministry of Magic keeps tabs on people apparating. That's why you have to have a license to do it, and the moment you abuse it you can find yourself in serious trouble (or Azkaban!).
-- J.K. Rowling, when asked why wizards don't Apparate into banks and steal things (Sch2)
Ministry of Magic licenses are issued for a number of activities which are deemed dangerous, which could be misused for nefarious purposes, or which require special training or actions to keep the wizarding world hidden from the Muggles. These include:
- Ownership of Crups (FB, JKR-W1)
- Sale of magical artifacts (DH11, JKR-W1)
- Apparition (Sch2, HBP18)
- Producing Floo Powder (only the Floo Pow company is so licensed) (Pm)
In the United States, ownership of a wand requires a license from MACUSA (WFT).
Ron Weasley actually applied for and obtained a license to drive a car, although he had to Confund the examiner in order to pass (DH/e).
Interestingly, there doesn't seem to be any license required to fly on a broomstick but there is an office in the Ministry of Magic called the Broomstick Regulatory Control, part of the Department for Magical Transportation (OP7).