Classicism, in the arts, refers generally to a high regard for classical antiquity as setting standards for taste which the classicist seeks to emulate. Classicism is usually contrasted with romanticism; the art of classicism typically seeks to be formal, restrained, and Apollonian rather than Dionysiac, in Friedrich Nietzsche's well known opposition.
Unity of time referred to the need for the entire action of the play to take place in a fictional 24-hour period
Unity of place was as it sounds, the action had to unfold in a single location
Unity of action meant that the play should be constructed around a single 'plot-line', such as a tragic love affair or a conflict between honour and duty.
The language also had to be of the most exalted kind, excluding 'low-life' characters and smutty jokes.
Classicists did not approve of Shakespeare, who broke all these rules and plenty more.
This embodies all art forms, the sciences, and philosophies.
When cultural movements go through revolutions from one to the next, genres tend to get attacked and mixed up, and often new genres are generated and old ones fade.
Originating in the 19th century with Symbolism, the Modernist movement composed itself of a wide range of 'isms' that ran in constrast to Realism and that sought out the underlying fundamentals of art and philosophy.