Occasional Flyer, the answers so far have all been along the lines of "basically everything" to "look in the ANO" or "look in the classifieds". All true, but since you seem to be new to this whole flying thing, let me put it in simple words.
The basic PPL, what almost everybody goes for, legally allows you to fly "single engine piston" aircraft up to a max. take-off weight (well, mass actually) of something like 5700 kilograms (I think - have to look up Air Law for the exact number). Initially, the aircraft also cannot be "high performance" (over 250 Hp), nor "complex" (retractable gear, variable pitch prop) nor a "taildragger".
This includes basically all your typical school/rental two and four seater, single engine aircraft like Cessnas, Pipers, Robins and so forth.
Onto the basic PPL you can add endorsements, qualifications and ratings for different other aircraft types, configurations and characteristics, such as complex, high performance or taildraggers, or aircraft that have multiple engines, aircraft that are heavier than the mass limit mentioned, aircraft with jet or turboprop engines, complex aircraft with retractable gear, and so forth and so on.
And indeed, technically/legally you can obtain an A380 type rating on just a PPL. Although typically those type rating courses teach the type rating to ATPL standards and as a 45-hour PPL you'll have to struggle to keep up with your classmates....
That's the legal side. From a practical point of view, virtually all operators/owners of aircraft require you to do a checkout with a designated instructor (or check pilot) before they allow you to fly their precious airplane(s). That's in addition to all the legal requirements and is just a way to make sure for the operator (and sometimes the insurance as well) that you indeed know enough about the aircraft to fly and handle it safely.
Having said that, if you are competent and experienced on type, a typical checkout will take no more than 30 to 60 minutes and is sometimes combined with things like local area familiarization as well. If you're not familiar with the exact type or model, the checkout might be a little longer to much longer, depending on how fast you pick things up, your experience with different subsystems and so forth.