Any single engined piston aircraft provided it has a tricycle gear, i.e. a nose wheel not a tail wheel; a fixed pitch prop and fixed undercarriage. To fly a tail wheel, variable pitch propellor aircraft or one with retractable gear you need 'differences training' which has to be signed off in your log book.
You can only fly in daylight, in sight of the surface and clear of cloud (the actual rules are a bit more complex than this and depend on the airspace you're flying in). You can't do instrument approaches unless you have at least an IMC rating. To fly at night you need a night rating. You can carry passengers provided you have done 3 take-offs and landings in the previous 90 days as commander of the aircraft. You are currently limited to one passenger if you are flying an aircraft with a Permit to Fly rather than a full Certificate of Airworthiness (you won't be doing your training on a Permit aircraft). Many older aircraft are flown on a Permit as well as home built aircraft. They are also restricted in that you can't fly over built up areas in them, or fly them at night or on instruments, even if you have an IMC rating.
Hope this helps, and good luck with the training
