Activation energy is the amount of energy that the reactants need in order to react and become the products. For example, when burning natural gas (also called methane, or CH4), you need a spark to light it. This spark is hot enough to start the reaction off: CH4 + 2O2 --> CO2 + 2H2O. Here, the reactants are CH4 and O2, and the products are CO2 + 2H2O. This reaction is exothermic, meaning it gives off heat. This heat can in turn provide the activation energy for more molecules to react together, so this reaction will keep on going until you run out of one of the reactants.
A catalyst is a substance that speeds up the rate of a reaction, without being used up in the process. It speeds up the reaction by providing a reaction pathway with a lower activation energy, meaning that more molecules are able to react. This is very useful in industry, because you can make your product more quickly, and you have to heat it up less, making the process cheaper.