The piano evolved from the harpsichord in the early 1700s. In the mid 1800s, the instrument went through many changes (tone, range, pedals etc) and became similar to the piano we know and love today.
The role of the piano in classical music evolved over time as the instrument itself did. Solo pieces became longer, more complex and technically demanding. The piano has been used with every combination of instruments imaginable, and featured in many well known concertos during the Romantic period. Here, the piano is featured as a solo instrument accompanied by a full orchestra. Beethoven, Tchaikovsky, Rachmaninov and Grieg all wrote notable romantic piano concertos you should check out.
In regards to the specific terms you addressed, here is a brief overview of each:
A romantic character piece for piano written in a melancholy, somber style. Often consists of an expressive melody played over a broken, moving chord pattern. Chopin took the idea from an irishman, John Field, who is credited with writing the first nocturnes.
Etude literally means "study" in French. They are designed to address a specific area of technique (perhaps a certain scale or arpeggio) in an instrument, and are used as exercises for practicing. Over time, various composer have written extremely impressive Etudes for a variety of instruments that often make their way into concert programs.
A stately Polish dance that is thought to have originated for its use in court ceremonies and processions. Bach, Beethoven and Schubert all experimented with the Polonaise. However, it was Chopin who really brought the style into a state of heroic nationalism.
Hope that helps!