From the late 17th century, the term concerto has been used to signify a composition showcasing a solo instrument, or possibly a selection of instruments, against the backdrop of whatever standard-issue orchestra the composer's musical tradition calls for (initially string instruments, but Classical and Romantic taste upped the ante to almost any size multi-section orchestra). A short, three-movement piece is the norm, with a slower, more reflective middle part (andante, maybe largo) sandwiched between allegros of one sort or another.
Baroque composers used the concerto as a medium to display their own prowess and that of their favourite performers, but also of course to take the instrument or instruments in question to the extremes of their technical and expressive capabilities. Antonio Vivaldi, a prolific and proficient source of such concertos, took this idea to the edge. When he took an interest in some instrument (for instance the bassoon, much favoured in Venice were he was employed for several years), he could produce literally dozens of three-part concertos, each one exploring, perhaps, the heights and depths of tone achievable, or the different voices and qualities of sound which could be produced. Mozart's concertos (say the ones for horn or clarinet) are really all about one instrument, and a light, clever theme to match some particular timbre or colour unique to that instrument.
By all accounts, these five- to ten-minute snippets of musical deftness were massively popular entertainment, and many 18th-dentury composers earned their keep churning them out. Their overriding attraction must be their clarity: one or two distinct voices, against a neutral orchestral background, playing a single theme, often reiterated in clever disguises, or perhaps balancing two very simple motifs.