The concerto was originally a work for one or more voices with instrumental accompaniment. During the 17th century it evolved into a form where a single soloist would play with an instrumental ensemble (concerto); or a group of instrumentalists would play with the larger ensemble (concerto grosso) in multi-movement orchestral works which featured passages or entire movements designated to be played by solo instruments. In its form, a concerto is a sonata with certain modifications.
1st movement - Sonata form (Exposition, Development, Recapitulation - fast)
2nd movement - Song form, Rondo form or Variation form. Slow
3rd movement - Rondo. Fast
The Minuet and Trio are omitted from the concerto.
Solo instrument and orchestra develop theme through various keys relating to the tonic.
The modern concerto was developed during the Baroque period under the influence of masters such as Antonio Vivaldi and Johann Sebastian Bach. Vivaldi developed a form dubbed the "Italian Form," which standardized the concerto into a three-movement work featuring a fast introduction movement, a slow second movement, and a final quick, dance-like movement. Vivaldi's prolific output included over 250 concerti, such as The Four Seasons (a series of four concerti featuring a solo violinist), and many other works featuring solo flute and guitar/lute. The "concerto grosso" form was developed in the late 1700s by Archangelo Corelli and realised its Baroque potential in the opus 3 and opus 6 sets by G.F. Handel
The concerto underwent a process of formalization and development during the Classical period, particularly by Wolfgang Amadeus Mozart. A virtuosic musician in addition to composer, Mozart wrote many concerti for a wide variety of instruments, including the clarinet, violin, and piano. In contrast to earlier concerti from the Baroque period, Mozart provided a short solo section for a musician to "show off," called a cadenza. Cadenzas would typically occur after the recapitulation and second development of the first movement (Mozart adopted the sonata form for many individual movements of his concerti), and would be left to the improvisational abilities of the musician. It was understood, however, that the cadenza would refer to and likely further develop thematic material from the actual movement. Mozart did leave a few written cadenzas for his piano concerti.
During the Romantic period, the concerto gained much more notoriety due to the introduction of bona-fide virtuosi. The concerto became one of the highest forms of classical music, and was often a work to be commissioned by individual performers. For example, Joseph Joachim, a famous violinist of the 1800s, commissioned Johannes Brahms to compose his violin concerto, one of the most highly regarded pieces in the violin repertoire. There was a resurgence of concerto grossi, but typically for smaller groups (no more than two soloists).
During this period, the concerto underwent a massive shift in form and focus. The double exposition of classical concerti fell by the wayside, and many (such as Mendelssohn's violin concerto) bring in the soloist at the very beginning. In addition, the form tended more and more to the symphonic, although the virtuosity of the solo part continued to increase as well. This is perhaps best exemplified by the growth of the cadenza.
Composers begin writing their own, multi-page cadenza, allowing it to serve a functional purpose in the music. Oftentimes, composers would work with whoever commissioned the concerto when writing the cadenza, either to compose to their individual playing styles or to determine what was practical and what was not. The great Romantic concerti - examples of which are Antonin Dvorak's Cello Concerto; Felix Mendelssohn, Brahms and Tchaikovsky's Violin Concertos - feature elaborate cadenzas which only the most advanced musicians can attempt to play due to their virtuosity.
Today, the concerto lives as one of the most often-performed styles of music. Many musicians still re-record the same concerti that have been recorded for nearly eight decades, although a practice of composing an individual, unique cadenza has become popular, particularly with violinists such as Joshua Bell. Many of the classical concertos have become standard solos for music students: nearly every Horn player, for example, has learned the Mozart works at some point.