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There were two principal types of concerti in the Baroque period.  The first type that composers created was the Concerto Grosso.  This type evolved from the earliest form of stand alone instrumental  music- the sonata. Composers in the early 17th century rather quickly began to add additional instruments to their solo sonatas, presumably to delight and surprise their royal, or at least noble patrons.  As these additional instruments were mixed together, composers, perhaps encouraged by the instrumentalists themselves or simply the need to do something new, began to "pit" these instruments against each other in various ways and the nascent concerto grosso was born,  Arcangelo Corelli in the latter quarter of the 17th century developed the form to an early  pinnacle and influenced many composers thereafter.  Two of his adherents were Georg Fredric Handel from Saxony and Johann Sebastian Bach in Thuringia.  They pushed the form to the highest levels in the 18th century,

The origins of the solo concerto are clearer and two names figure prominently; Giuseppe Torelli and Antonio Vivaldi.  Torelli generally gets the nod for inventing the solo concerto around the year 1700 or so although he also wrote concerti grossi as well.  Torelli was a violinist and violist yet his most famous concerti were written for trumpet.  He composed more than 30 concerti for from 1 to 4 solo trumpets in the early 1700s.  Antonio Vivaldi made the real breakthrough with the popularization of the single instrument concerto with the publication of his L'estro Harmonico (literally harmonic heat!) Opus 3 by the renowned publisher Etienne in Amsterdam in 1711.  This set of 12 concerti should be considered one of the most important musical events in history as it provided both wonderful works on their own musical merit, but more importantly provided a new way of thinking about instrumental music to composers all over Europe.  Two eager recipients of this collection of solo concerti were JS Bach and GP Telemann.