The concerto is usually a major work written for a single solo instrument backed by a full orchestra. The word derives from Latin 'concertare' = to work together. In early concertos, the performers definitely worked together all the time. In later works, the soloist emerges as a virtuoso, with substantial solo passages.
There are usually 3 movements, the first is in sonata form, like the first movement of a symphony, except that near the end, the soloist plays a cadenza, which is a sparkling display of virtuosity. The second movement is slow, also as in the symphony. There is no minuet and trio, which is the usual form for the 3rd movement of a symphony. Instead, the 3rd and last movement is fast, often in rondo form.
Mozart wrote many piano concertos, one of the most well-loved is no 21 in C. The slow movement of this has a wonderful melody, which was used as background music for the film 'Elvira Madigan'. The film is more or less forgotten today, but this modern sub-title has stuck to Mozart's 21st piano concerto. Another unforgettable piano concerto is Grieg's in A minor, with its very dramatic opening passage on the piano. Beethoven's 5th piano Concerto, nicknamed the 'Emperor', is magnificent, as also is Tchaikovsky's in Bb minor.
World famous violin concertos include Beethoven's in D, Mendelssohn's in E minor, Bruch's in G minor, and the Brahms and Tchaikovsky violin concertos, both in D. Mozart's horn, flute/harp, and clarinet concertos are superb, as also are Haydn's trumpet concerto and Rodrigo's guitar concerto.