Concerto
The musical term "Concerto" derives from the Italian term "Concertare," meaning "to compete" or "to fight." In broadest definition, a concerto is an instrumental piece of music using the contrast of two (or possibly more) groups of musicians. Throughout history, that has come to mean several different things.
Baroque era
The Baroque concerto divided the orchestra into two groups: the ripieno and the concertare. The ripieno was the larger of the two groups, encompassing any who were not in the concertare - which could be as small as a single instrumentalist. Distinctions were made along the lines of the "solo concerto" (a concerto for a single player against an orchestra) and the "concerto grosso" (a concerto for a group playing against the remainder of the ensemble) though structurally and musically there was little difference between the two. There was no fixed format, though there was a marked preference for using a ritornello, a refrain-like musical passage for the ripieno which punctuates the music, framing the concertare's statements and occasionally providing thematic material for the smaller group. This practice continued into the Classical and Romantic eras of music.
Classical era
The death of Bach and the rise of the Stile Galant also saw a marked decline in the concerto grosso. Few concerto-style pieces were written with a group of instrumentalists in mind; the composers preferred to highlight a soloist's efforts. Wolfgang Amadeus Mozart wrote at least twenty-one concertos for solo piano, most first performed with himself as the soloist. The Classical era added a new stylistic aspect to the concerto, that of the cadenza, an improvised (or quasi-improvised) moment for the soloist to demonstrate their level of skill and talent. The cadenza had existed for decades in Opera, but had until this point been unknown to instrumental music. Ironically, as the cadenza gained greater prominence in the concerto, its use in opera died down with the operatic reforms of Cristoph Willibald Gluck.
Some very few examples of conterti grosso of the Classical era do exist: Mozart wrote a "Sinfonia Concertante" for violin and viola soloists; Ludwig von Beethoven composed a "Triple Concerto," with solo parts for violin, 'cello, and piano; Felix Mendelssohn and Mozart both wrote concertos for two pianos.
Romantic era
After Beethoven, the Romantic-era composers took the glorification of the soloist a step further, writing solo parts which were considered to be all but impossible. Franz Liszt's and Sergei Rachmaninov's piano concerti are virtuosic in the extreme, as are Niccolo Paganini's violin concerti. These works became showpieces for the soloist, with the orchestral function greatly reduced and sometimes almost non-existent - Frederic Chopin's two concerti, for example, can be amply accompanied by a string quartet instead of a full orchestra. In fact, where the idea in the past had been to create a dialogue of equals, almost a partnership between ripieno and concertare, in the Romantic era the focus was entirely on the soloist, reducing the orchestral role to that of accompanist. The concept of the ritornello nearly disappeared. One notable exception occurs in the four concerted works of Johannes Brahms - his two piano concerti, his violin concerto and his "Double Concerto" for violin and 'cello (one of a handful of Romantic concerti to include more than one soloist) - where the orchestra is often the equal and occasionally the leader in the music. Brahms' first piano concerto was criticised as a "Symphony with piano" for the equitable treatment of both groups and the long orchestral lead-in.
The modern concerto
As music has progressed and orchestral commissions have slowed, composers have been producing fewer and fewer concerti since the turn of the twentieth century. Since the end of the Romantic era, most successful composers have not themselves been virtuosic instrumentalists, and the symphony orchestra has become steadily less amenable to the performance of new music. The concerto itself, though, hasn't changed a great deal, though some composers have tried using the cadenza in different ways - Dmitri Shostakovitch makes the cadenza a complete movement of its own in his two 'cello concerti - or eliminating it altogether, as Alban Berg did in his violin concerto.
Works for soloist(s) and orchestra
There have been several works written for soloist and orchestra that aren't labelled as concerti but still feature the contrast in sound and texture between a small group and a large one. Beethoven wrote two single-movement works for solo violin and orchestra labelled "Romances," for example, and César Franck's only work for solo piano and orchestra is his "Variations Symphonique." These are often referred to as "Concerted" works. Robert Schumann wrote several such pieces for orchestra and solo piano; one (a "Fantasia") he expanded into a full-scale three-movement concerto.