Neoclassicism was a school of thought that developed in the 18th century. Its main attributes were reason, order, and serenity. They wanted to move away from the extravagant and over-the-top Baroque period that had come before and move into more orderly and rational styles of art. They took great inspiration from the Classical period of Greece, hence the name Neoclassicism. They disparaged the Medieval way of art as barbaric and something not to be replicated. Neoclassical architecture is common in the United States, such as the Capitol Building and the First Bank.
In this painting, by one of the most famous and influential painters of the Neoclassical style, Jacques Louis David, you can see some of the key elements of Neoclassicism. The emphasis is on horizontal and vertical lines with the subject in the center. The subject has order and composure, and there is no visible emotion.
To contrast Neoclassicism is Romanticisim. Romanticism gained influence in the early nineteenth century. Romanticism was not against reason but said that life should not only be made up of reason. There was an emphasis on spirituality, sometimes through traditional ways but often through nature, and through nature, you would get a feeling that God was the creator. Romanticists would have an interest in things far away, either in the past or geographically. The Romanticists celebrated Gothic architecture as seen in the Houses of Parliament in London.
In this painting, you can see that the emphasis is not on straight lines but more on emotion and feeling. The landscape has a feeling of mystery about it. Some Romantic writers include Mary Shelley, author of Frankenstein, and Victor Hugo, author of Les Miserables. In these books, the emphasis is not on order but more on the way the characters feel.