The flying carpet, also known as a magic carpet, is a form of magical transportation used by some nations of the wizarding world. Flying carpets are particularly popular in Asia and the Middle East, with India, Pakistan, Bangladesh, Iran and Mongolia among the main nations using them. The Ministry of Magic in Great Britain has put a ban on all Flying Carpets because they are classified as Muggle Objects.
Flying carpets were once an accepted form of travel for Britain's magical community, but they are now banned due to being defined as a Muggle Artefact by the Registry of Proscribed Charmable Objects. It is therefore now against wizarding law to charm carpets or fly them there, although they are still legal in other countries. Arthur Weasley was very much involved in the introduction of this legislation due to his position in the Misuse of Muggle Artefacts Office. It is known that the ban was relatively recent, not only from Arthur's involvement, but also from the fact that Barty Crouch Sr.'s grandfather owned a 12-seater Axminster back in the days before they were prohibited.
The obvious advantage of the carpet over a broomstick is that they can seat a number of people, and are no doubt more comfortable to ride. It's still likely to be pretty chilly traveling on one, however, as like a broomstick, they offer no protection from the elements.
A merchant by the name of Ali Bashir was very keen to export flying carpets to Great Britain and was very upset that local laws prevented him from doing so. He regularly berated Arthur about the subject but it is very unlikely that the law will ever be changed.