Regulations on short-term rental properties tend to come at the local level rather than from a national government. I don't think a place exists where a renter would have to apply for a license to stay somewhere, but in most countries in Europe now there are cities or regions that require Airbnb hosts to register as businesses with a local authority. Airbnb doesn't necessarily enforce the rules in every location - logistically, that would be impossible - so a guest will encounter listings that are not technically legal.
In some places, all nightly accommodations (including homestays) are supposed to be registered; in others, only whole-apartment listings are subject to restrictions or bans. Taxing is an altogether separate issue that gets more complicated, as hosting revenue can be considered both income (like employment) and also subject to a local tourism tax.
Whether you're a guest or host, my best suggestion would be to research the laws in the specific city or district or island that you have in mind. Be prepared for some confusing information - oftentimes, you'll run up against laws that pre-date Airbnb and a lot of ambiguity about how they apply to different kinds of accommodation. You'll also find that in most locations, the average host isn't playing by the rules as they appear on paper.
