A catalyst is an agent that speeds up a chemical reaction, whilst not being consumed in the overall reaction.
For example, the reaction A + B → C might be very slow, only producing one 'C' every 20 minutes. However, if we add D to the reaction and it subsequently speeds up, D would be a catalyst, providing that there is still the same amount of D in our beaker as when we added it.
Catalysts work by lowering the activation energy of a reaction, but this is such an important topic that it'll have to be dealt with later. I hope you'll excuse the "phrase-dropping"!
I was always taught that an enzyme is a biological catalyst, and that's still the definition favoured by many authors. However, some authorities insist on limiting the term to (biological) catalytic proteins, and thus distinguish them from things like ribozymes, which are catalysts that happen to be made of RNA. As usual, I don't have much patience with definition games, so ask your lecturer which definition he prefers, and give him that one back come exams.