The concerto was established as a form of composition in the Baroque period. Starting from a form called Concerto grosso introduced by Arcangelo Corelli, it evolved into the form we understand today as peformance of a soloist with/against an orchestra.
The main composers of concerti of the baroque were: Tommaso Albinoni, Antonio Vivaldi, Georg Philipp Telemann, Johann Sebastian Bach, George Frideric Handel, Pietro Locatelli, Giuseppe Tartini, Francesco Geminiani and Johann Joachim Quantz. The concerto was intended as a composition typical of the italian style of the time and all the composers were studying how to compose in the Italian fashion (all'italiana).
The piano concert, with the exception of the organ and some harpiscord concertos by Johann Sebastian Bach, was not yet a common practise in the baroque period simply because the piano did not exist yet. The keyboard concerts became more and more important when evolving from harpsichord, to fortepiano, and in the end to the the modern piano. The increased volume and the richer sound of the instrument allowed the keyboard instrument to better compete with the full orchestra.