This is the question people most often arrive here searching for, so it seems the right place to start the forum.
The short answer for a house in England is that it can be permitted development, and the trap is the one almost nobody mentions: the rules only allow a unit that is not used solely for cooling. A reversible unit, one that heats as well as cools, can be permitted development. A cooling-only air conditioner cannot, at any size, in any position.
The rest of the conditions are about where it goes and how big it is, and the planning checker walks through them one at a time and tells you which condition you fall foul of rather than just saying no.
Two things it deliberately does not cover: it is England only, and it assumes a house rather than a flat. Wales, Scotland and Northern Ireland have their own regimes and are not the same.
If you have been quoted for a cooling-only unit and told it needs no permission, that is worth going back to the installer about.