The concerto emerged as a distinct musical form in the late Baroque period.
A concerto is work for a soloist and an orchestra. Hence, Concerto for violin and orchestra, Concerto for piano and orchestra etc.
In a concerto the solo instrument and the orchestra have a relationship in which dialogue between the two is the glue which binds it all together. Thematic ideas are passed about, exchanged and developed, and virtuosic solo passages are supported and punctuated by the orchestra,
There are normally three movements in a concerto - the 1st is usually in a slightly modified Sonata form with an exposition, development and recapitulation but with a bit more complexity due to the additional role of the soloist. The 2nd movement is usually a slow, lyrical movement and often is either in Song form (ABA) or in Variation form. The 3rd movt, usually fast and lighter and more joyful in mood, is usually either in Sonata form or a Rondo (ABACAD etc.).
The concerto is also a platform for the soloist to demonstrate their virtuosity on their instrument, so there is usually a place very near the end of the 1st movemnt for a Cadenza - this is a quasi improvised space (although composers often provide one) during which ideas from the themes are developed and played around with. Fast passages with scales and arpeggios are common.
In some concertos there are cadenzas in the 2nd and 3rd movements too, but these tend to be on a smaller scale, both in terms of length and complexity.