A catalyst speeds up a reaction and comes out unchanged at the end, as you have seen. It works by giving the reaction a different route, with a lower energy hill.
With a lower hill, many more collisions have enough energy to get over, so the reaction goes faster at the same temperature. The catalyst takes part along the way: the reactants hold on to it, react, and let go, and the catalyst is left as it was, ready for the next molecules. A small amount of catalyst can speed up the reaction of a great many molecules.
Hydrogen peroxide and a black powder
Hydrogen peroxide, H₂O₂, has one more oxygen atom than water, and its two oxygen atoms are joined by a single bond. It breaks down into water and oxygen gas. On its own it does this so slowly that a bottle from the pharmacy keeps for months. It comes in an opaque bottle, because light speeds the breakdown.
Add a pinch of manganese dioxide, MnO₂, a black powder, and the hydrogen peroxide fizzes at once with bubbles of oxygen. When the fizzing stops, the powder can be filtered out, dried and weighed, and it weighs the same as before. It is a catalyst, and it can start the next batch fizzing too.
In a car's exhaust
A petrol engine's exhaust carries carbon monoxide, the poisonous gas that burning makes when oxygen runs short, along with unburned fuel and nitrogen oxides, which help make smog. A catalytic converter in the exhaust pipe cleans them up. Inside is a ceramic honeycomb coated with a thin layer of platinum, palladium and rhodium, and its thousands of narrow channels give the gases a huge surface of metal to touch.
On the metal, carbon monoxide reacts with oxygen to make carbon dioxide: 2 CO + O₂ → 2 CO₂. Unburned fuel burns to carbon dioxide and water, and nitrogen oxides turn into nitrogen gas, which makes up most of the air. The converter works once the exhaust has heated it to a few hundred degrees Celsius.