The C clarinet is a few inches shorter than the B-flat. It will sound one
whole tone step higher, and is pitched in the same key as violines and piano.
If you play very often together with piano or guitar (like in bands),
it is very convenient to use a C clarinet. Since it is smaller it it also
used in musical schools to educate children. In symphony orchestras you will
rarely see the C clarinet even though some pieces are written in C (by for instance Brahms and Beethoven).
The clarinettists are usually so skilled that they transpose directly on their B-flat instrument.