A clarinet concerto is a piece of music written for clarinet to be played with an orchestra. The most famous clarinet concerto ever written is Mozart's Clarinet Concerto in A. Attached is a related link of the youtube recording of the piece by Martin Frost and the wikipedia article about the Mozart Clarinet Concerto.
A clarinet quintet is a piece for clarinet and string quartet. A string quartet typically has 2 violins, a viola, and a cello. There are numerous composers that wrote for this genera, but since we talked about the Mozart Clarinet Concerto above it is interesting to note that Mozart also wrote a Clarinet Quintet. This piece is also attached.
Hope this helps! Good luck with the clarinet.