styles of paly (Realism, classicim, romanticism)?

Answer 1

Neo-classical drama had its heighday in France during the 1700s. It was heavily influenced by Greek classical drama and tried to apply the "rules" of the unities. This limited the dramatic form.
Romanticism, mainly during the 1800s, followed partially as a reaction to Neo-classicism. It released itself from the classical limitations but held on to some concepts such that good and evil are distinct and good is eventually rewarded. Melodrama was the most popular form of Romanticism and used a lot of spectacular stories and settings.
Realism, primarily during the 1900s, brought the idea that things are not black and white but many shades of grey. Many things were explored including disease, poverty, social prejudice, etc. Realism brought in more realistic settings and made the director more important as "blocking" became much more necessary.

Answer 2

There are a wide and wild variety of styles.

The first three listed are essentially realistic, the others are more abstract. This is by no means a complete list, but covers some main plays in the last couple of centuries.

Naturalism: social and economic forces act on characters.

Realism: psychological motives of chracters; verisimilitude

Romanticism: emphasis on the emotion and the "harmony of opposites"

Expressionism: emphasizes inner experience and reaction rather than solely realistic portrayal

Constructivism: art for the industrial age, with machine-like movement

Absurdism: theatre that finds any attempt at understanding the world as futile