A carburetor (commonly called just "carb") is a mechanical device that mixes fuel and air, for combustion by an engine. They have needles to adjust the amount of fuel that is mixed into the air, a throttle to change the amount of fuel/air mixture into the engine to change the engine speed, and sometimes on gasoline engines, a choke valve to restrict the amount of air going into the engine for easier starting.
A slide carburetor has a throttle valve that slides out the side. This reduces air restriction, for better performance, and requires the servo to move less to open it fully, for better throttle response. They also require a more precise alignment of the carburetor and throttle servo to prevent binding, which can result in dangerous stuck throttle conditions, which can often only be resolved by killing the engine after the vehicle is stopped by a large immobile object like a tree. Slide carbs are used on all big block nitro engines, and on many small block nitro engines. They are fairly prone to sticking because of dirt, and on land vehicles, an air filter is always used.
Rotary carburetors have a barrel with a hole in it that rotates. This takes more servo movement to completely open, and reduces throttle resopnse. They also have more restriction, which can reduce overall power. Some people feel that these are better for offroad use, because they have more throttle range. They aren't used on big block engines. They are as prone to dirt as a slide carb.
Butterfly carburetors are common on gasoline RC engines and small 2 stroke engines designed for other purposes but used on RCs. They have a butterfly, or a thin piece of metal that rotates to open the throttle. They don't stick when they get dirty, but the dirt can still cause engine damage and poor performance. These carburetors are also used in alot of full scale applications, like cars and lawnmowers. They often have a second butterfly as a choke valve, to reduce air flow, temporarily richening the mixture for easier starting. These carburetors have more restriction then other carburetors, but the engines they are used on are larger, and the performance doesn't matter as much.