A cat uses a coating typically made of platinum or palladium/rhodium to act as a chemical catalyst (hence the name) to change carbon monoxide (CO) and unburned hydrocarbons (HC's consisting of mostly fuel and oil) to carbon dioxide (CO2) and water (H2O) along with oxides of nitrogen (NOx) being converted to nitrogen (N2) and oxygen (O2).
The cat does this by the process of either chemical oxidation or reduction while passing the exhaust gas over a core (as in the photo below) covered with the chemical catalyst.
These chemical reactions require the exhaust gas (and the catalyst) to be fairly hot in order to work well, thus the need for secondary air injection during the cold start phase of engine operation to keep emissions low until the exhaust and catalyst warm up.
Eventually, the catalyst wears down and the cat doesn't work as well. Emissions go up and the CEL is turned on. It is generally not possible to replace the chemical catalyst because it is deposited on the core during the manufacturing process.
Platinum is fairly rare and because of its rarity, only several hundred tons are produced per year. As such, platinum is often more expensive per ounce than gold.