If you ask most of the public what a catalyst is they will probably tell you that it has something to do with reducing pollution from cars. Whilst catalytic converters are important they are just one of the many applications that catalysts are put to today.
Catalysts do one very simple but vitally important thing, they speed up chemical reactions without being used up in the process. The way catalysts achieve this is by providing an alternative route for the reaction to follow that has a lower activation energy.
You can think of this as being on one side of a mountain pass. Even though your destination on the other side is closer to sea level you just don’t have enough stamina to climb the pass in order to go back down the other side. The catalyst provides a tunnel that takes you through the mountains so that you never have to climb that high. An important thing to remember is that the route takes two-way traffic and so it cannot affect the final concentrations of reactants and products, just how soon the final concentrations are reached. The normal way of saying this is that a catalyst cannot affect the equilibrium position of a reaction, just the rate.
The two types of catalyst you will meet are called homogeneous and heterogeneous catalysts. Whilst they achieve the same result they work in fundamentally different ways.
With homogeneous catalysts the catalyst and all the reactants are in a single phase usually either liquid or an aqueous solution. Invariably the catalyst is taken up into the chemical reaction before being ‘spat out’ at the end.

The following example is a real, though economically valueless, catalytic reaction. It is helpful in understanding how catalytic reactions can occur.
Persulphate ions (S2O82-) are powerful oxidants and iodide ions (I-) are easily oxidised. However in an aqueous solution the reaction runs very slowly; it should be easy to see why:
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If we add a small quantity of iron ions, say Fe2+ then the reaction has another route.

If we had started with Fe3+ ions then the reaction would simply have taken the opposite order (not the reverse reaction though).

The important points are that the same overall reaction is achieved and that the catalyst is returned unaltered at the end of the reaction.
Why should the introduction of Fe2+ or Fe3+ ions speed up the reaction?
What other stage should be introduced in order to exactly duplicate the uncatalysed reaction?
Most industrial catalysed reactions are of the heterogeneous type. Here the reactants and the catalyst are in different phases. Usually the catalyst is in the solid phase whilst the reactants are either liquid or gaseous. In industrial applications the catalyst is often supported on a substrate that allows the effective surface area to be increased and the catalyst to be fixed in place.
The route by which an heterogeneous catalyst works is as follows:

Firstly the reactants are adsorbed on to the surface of the catalyst. This is a chemical reaction as there is an interaction between the electrons of the reactants and the atoms on the surface of the catalyst.
(Remember that adsorption and absorption are different. In adsorption a molecule binds to the surface of the material whilst in absorption it is taken in to the body of the material.)
Secondly the adsorbed reactants (particularly the lighter ones such as hydrogen) are free to migrate over the surface of the catalyst.
Thirdly, when the reactants meet they are free to react but are still bound to the surface.
Finally the products of the reaction are desorbed from the surface allowing them to move away and freeing up catalytic surface area for further reactions.
This points up that a good catalyst should bind moderately well to the intended reactants and products.
The classic examples usually used in education are the Haber process for the production of ammonia and the contact process used to produce sulphuric acid.
So what makes for a good catalyst and how can you develop one? There are a few general rules and a lot of questions:
The catalyst should have a large surface area.
The catalyst should have a variable coordination number, this defines the number of nearest neighbours an atom or molecule can bind to. Having a variable number allows potential catalyst to link to extra molecules at the surface when reactants are adsorbed and to release them again when the reaction is complete.
The catalyst should have a variable oxidation state, this is the number of electrons an element can donate or receive. By taking part in electron exchange a potential catalyst can initiate chemical changes in molecules that bind to its surface. The electron exchange can also promote the binding of the molecules to its surface in the first place.
The catalyst should be able to vary its stereochemistry, this defines the angles between bonds and so affects the shapes of molecules that can be created.
A good catalyst will possess some or possibly all of these characteristics.
What benefit does a large surface area grant to an heterogeneous catalyst?
To put this advise in context the Haber process has been in use for around a century but the description of the detailed reactions in the catalytic process was the main reason for the awarding of the 2007 Nobel Prize for Chemistry. Much of the time we can work out what might make a good catalyst but developing an excellent catalyst for a particular reaction involves a lot of directed trial and error. A recent professional review of an international catalytic research project recommended that the catalysts that produced poor results should not just be ignored but should be studied in more detail; if we knew precisely what made some catalysts poor performers we might be better able to design new high performance ones far more easily.
You can think of a promoter as being “a catalyst’s catalyst” Whilst not a catalyst in its own right a promoter speeds up an already catalysed reaction without itself being used up. They are of particular interest in commercial organic chemistry. The reason for this is that a reaction involving a complex organic molecule may well produce more than one potential product. Take as an example the hydrogenation of 1 - phenylethanol. Two possible reactions are shown here:

In fact both reactions will be catalysed. Now ethylbenzene can be processed to produce styrene, the monomer that is in turn used to produce polystyrene, the plastic that is used in disposable plastic drinking cups, ceiling tiles and scale model aircraft kits. If the company only intends to make polystyrene then the 1 – cyclohexylethanol is a waste product and an expensive one at that. They might try to find markets for the by-product but they would much prefer it if it were not made in the first place. Promoters allow them to go some way to doing this. By selecting the correct promoter the yield of ethylbenzene is increased.
Another small manufacturer may have specialised in producing 1 – cyclohexylethanol for a niche market. To this company the ethylbenzene is the by-product! Again a careful selection of a suitable promoter will increase the yield of the intended product and suppress the production of the by-product. This is not however something that a manufacturer does for itself, rather they will approach a larger company that specialise in the development and production of the catalysts that will then sell them a suitable catalyst already combined with its promoter.