The thickness of a cymbal is directly related to its power and speed. The thinner the cymbal, the faster and more precise it is. The heavier the cymbal, the louder and more cutting it is. In the big picture, what this means is that thin cymbals are perfect when you need swift response and precise sound with tight sustain. They're great cymbals for more laid-back musical genres, where the instruments are softer and a whole lot of penetration simply isn't needed to be heard over them. Thin cymbals are also right at home in studios and smaller venues, where a heavy cymbal would be overwhelming.
Crash and china cymbals make up the biggest share of the selections here, which makes sense: those types lend themselves best to the benefits of a thin design. There's plenty of variety, from the brilliant Sabian AA Holy China to the extremely musical Meinl Byzance Extra Dry Thin Crash. And those are just two examples from the dozens of thin cymbals you have to choose from. Like any instrument, it all comes down to what works best for you... so don't hesitate to dive in and investigate all the options - you'll be glad that you did!