The A Clarinet is the slightly larger sibling to the B-flat Clarinet. I have yet to see it used within a band setting. I do not know why. Every professional clarinetist will own an A Clarinet as it is essential for orchestral playing. The sound of the A is slightly darker and warmer than that of the B-flat. Some composers, such as Strauss, have used the B-flat and the A side-by-side in order to create interesting timbral and orchestration effects. I see no reason why this cannot be done in a band. As the instrument plays one half-step lower than the B-flat, it would be a logical choice to use it for the bottom part of the “B-flat” section. I could easily see a section of six players with the four upper parts on the B-flat instrument and the bottom two on the A instrument.
For information concerning using the B-flat and A clarinets and the differences between the two, read this article.
There also exists a so-called Basset Clarinet that is an A Clarinet with an extension down to a written C a major third lower than normal. This instrument exists solely for the purpose of the Mozart Clarinet Concerto. I’ve never understood why it has not been more widely used apart from its heavier weight. Only concert soloists will usually have a Basset Clarinet, so best to avoid any notes below the written E.
The most famous piece ever composed for a wind instrument: Mozart’s Clarinet Concerto performed on an A Basset Clarinet.