The simple answer is that hydraulics use a type of fluid, often oil, to drive some type of mechanism. Pneumatics on the other hand use compressed air.
When the systems are further anylyzed there are many other smaller differences. In hydraulics, fluid is kept in a tank which is non pressurized. The operator opens a valve and a pump forces the fluid from the tank through the valve and into the mechanism. Hydraulics are often used in high force applications where speed may not be of utmost concern.
In pneumatics, a compressor stores air in a pressurized tank. The operator opens a valve ( often solenoid driven) and pressurized air from the tank flows into the mechanism. Pneumatics are often faster than hydraulics but with a lower force capacity.