Let's take an ATPL pilot with a A320 type rating. As far as I know, this license will allow the holder to fly all A320 series jets, obviously. But what types of aircraft are implicitly included, so they can be flown without any additional license/type rating/mandatory training?
Single engine pistons? (I think yes)
Multi engine pistons? (Hey let's jump on that DC-6!)
I am looking forward to read your insights!
Since you didn't state where, under EASA rules (and it is by no means exhaustive, it is a big subject in itself):
Each type requires a class or type rating. A pilot can only fly what he holds a rating for. If he only has an A320 rating, he can only fly an A320.
A type rating is required for all turbine (jet and
turboprop) aircraft and aircraft with a weight limit over 5700 kg. Type ratings are generally for one entire type, (like all ATRs), but additional training can be required to fly certain variants (like the ATR -600 series).
Class ratings are more open and cover several types, like "all" single engine piston aircraft: Single engine pistons require a single engine piston [SEP] class rating, which comes in both land [SEP(L)] and sea [SEP(S)] variants. Multi engine pistons [MEP] also have a class rating, and also come in land [MEP(L)] and sea [MEP(S)] versions. There is additional training required for aircraft with tail wheels.
Helicopters are completely separate. ULs have historically been separate and required a light aircraft pilots license [LAPL], but authorities are slowly beginning to come round to accepting that ordinary pilots can fly some UL aircraft as well. You are now allowed to count UL hours as flying time on regular aircraft. Business jets count as any ordinary aircraft.
The pilot is also limited to flying aircraft registered where he is licensed. FAA pilots can fly US registered aircraft, pilots with EU licenses can fly in all aircraft registered in the EU. Many countries make validations easy for EU or US pilots to convert to, but the EU and US don't really recognize any other licenses without significant additional training.