What type of chemical reaction is it when sodium bicarbonate is added to citric acid?
The first step is usually called an acid-base reaction or a proton transfer. Any acid of similar strength to citric or acetic (vinegar) acids will react with bicarbonate this way, and it is very fast in water solution at room temperature. The products are carbonic acid and the sodium salt of the acid used--for example, sodium citrate. 

Carbonic acid is energetically unstable relative to carbon dioxide and water, to which it is rapidly converted via what most chemists would call an elimination reaction. I used to think that this occurs instantly, but carbonic acid is a species that actually exists in solution and can be observed and studied.

While there are many named organic reactions, I  agree with Ivan that there is no one universally agreed upon way to name them, so others may wish to weigh in with their opinions about this reaction.