So this is a good question; one easy answer is meteorites. Just in case you don't know, all meteorites are asteroids; they are called meteorites if they hit a planet. Many meteorites have hit Earth over the years, and some have been recovered, and studied.
But this doesn't cover all asteroids, and certainly not Ceres. A surprising amount of information is gleaned fro indirect study; e.g. by analysing orbits and sizes of an object one can work out its mass and hence its density. Which will narrow things down somewhat, other things that can be considered: albedo (the 'reflectiveness' of the object); formation methods (asteroids are widely though to be left over from planet formation, so if you know what planets started as you know what asteroids are). There is also spectrometry which is a bit tricky to explain if you don't have a technical background; similarly radar in this context.
This is an incomplete and basic list, but it's a start. Have a look here http://curious.astro.cornell.edu/about-us/72-our-solar-system/comets-meteors-and-asteroids/asteroids/295-how-are-asteroid-compositions-and-classifications-determined-intermediate for more information.