The concerto grosso is a form of baroque music usually having four to six movements and alternating between a small group of soloists (concertino) and full orchestra (ripieno).
The first major composer of concerti grossi was Arcangelo Corelli. After his death, a collection of 12 such pieces by him was published (presumably the movements were selected individually from a larger oeuvre) and soon spread like wildfire across Europe, finding many admirerers and imitators. Composers such as Francesco Geminiani and Giuseppe Torelli wrote concerti in the style of Corelli, and he also had a large influence on Antonio Vivaldi.
In Corelli's day, two distinct forms of concerto grosso were distinguished: the concerto da chiesa (church concert) and the concerto da camera (chamber concert). The former was more formal and generally just alternated largo or adagio movements with allegro movements, whereas the latter had more the character of a suite, being introduced by a preludio and incorporating many dance forms. These distinctions later became blurred and forgotten.
The most famous concerto by Corelli is No. 8 in G minor, the so-called Christmas Concerto, which ends with a furious allegro and then has an optional pastoral tacked on which should, in theory, only be played on Christmas Eve and must, in practice, often be played twice even when it isn't, due to its great popularity.
Corelli's concertino consisted of two violins and a cello, with a string orchestra serving as ripieno, both accompanied by a basso continuo. The latter was often realized on the organ in Corelli's day, especially in the case of the concerti da chiesa, but in modern recordings harpsichord realizations seem almost exclusive.
The other major composer of concerti grossi was Georg Friedrich Händel, who expanded the ripieno to include wind instruments. Several of the Brandenburg Concerti of Johann Sebastian Bach also loosely follow the concerto grosso form, notably the 2nd which has a concertino of recorder, piccolo trumpet, oboe and solo violin.