Catalysis is the study of catalysts. So, what is a catalyst? A catalyst is a material that speeds up the rate of an otherwise slow chemical reaction (activity). Another important function of a catalyst is to make sure the right chemical is produced (selectivity). In the process, the catalyst should not be changed (stability). So a catalyst should be active, selective, and stable.
Lets take an example. Fuel cells can convert hydrogen to electricity. The type of fuel cell being used in cars is made from a thin plastic membrane coated on both sides by a carbon powder. The carbon powder has nanometer sized platinum crystals all over the surface. Hydrogen and oxygen are fed to opposite sides of this membrane. Hydrogen gas (two hydrogen atoms bonded together) adsorbs on the platinum at the anode and is torn apart into two hydrogen ions (protons) and two electrons. The protons diffuse through the membrane while the electrons must go around in an external circuit. This provides the electricity. The electrons meet up with the protons and oxygen on the cathode side where the platinum helps combine these chemicals to form water. Hydrogen and oxygen very much want to combine into water. The platinum nanocrystals and design of the fuel cell make sure this happens in a controlled way that lets us capture electrical energy. The platinum speeds up the cathode reaction (active), minimizes formation of hydrogen peroxide (selective), and is resistant to aggregating into larger particles (stable). This is an example of electrocatalysis.
One more example. Very likely you actually sat right above a catalytic reactor on your way to school or work this morning. The catalytic converter in your car takes the carbon monoxide, nitrogen oxides, and unburned hydrocarbons leaving the engine and turns them into virtually harmless carbon dioxide and nitrogen. The catalyst is a ceramic "honeycomb" material with a silica-alumina powder washcoated onto the surface. The powder has nanocrystals of (guess what?) platinum and other metals all over the surface. This amazing catalyst destroys these pollutants as fast as the exhaust passes through the converter (active), minimizes formation of stinky byproducts like ammonia and hydrogen sulfide (selective), and remains active for years (stable). In fact, catalytic converters couldn't be used until the 1970's when tetraethyl lead was banned as an antiknock agent in gasoline because the catalyst is seriously poisoned by lead.
Catalysts are found everywhere we look, from manufacture of plastics, to enzymatic reactions in the body, to refining of petroleum to mention just a few applications. An excellent resource for learning more about catalysis is the North American Catalysis Society web page.