Romanticism: "Romanticism idealizes the remote, it glorifies the distant, both in time and place." writes J. Mordant Cook. Viewed this way, even the ever so formal looking architecture of the late 18th century can be recognized as romantic by its use of elements associated with the ancient republics of Rome and Greece. In the United States, the spirit of romanticism underlies much of the culture, including the architecture, from the dawn of the Republic through most of the 19th century. Hence it is important to realize that romanticism is not a style, but a pervasive force that impacts on many styles.
Academic Architecture: Often viewed as an antithesis to romanticism, this not entirely so. Sometimes it is used more superficially to refer to classical architecture. However, this can be confusing since as mentioned above, there was romantic use of classical elements. Like romanticism, the term academic is best understood not as style, but as a design methodology. Where romanticism emphasized imagination and observation, academic architecture emphasizes reason and math as ideals for design.
Academic is also associated with formal training, and generally that training specifically taught by the École de Beaux Arts in France. As a result, academic is both a process of design and a set of ideals. Academic architecture emphasizes the use of abstract reasoning and rigorous methodology. In most cases, academic designs favor symmetry, harmonies, and classical orders. However this is only when the ideals were strongly married to the methodology. Academic ideals at the end of the 19th century resulted in a divergence of expressions besides classical.
Picturesque: A term derived from mid-eighteenth century English landscapes that had been created to provide views befitting a picture. This became an ideal that swept into the neo-classical architecture of the time, and reached America at the end of the 18th-century. Its implementation was another matter, and at first was generally limited to the finest country houses. It remained an influential concept, particularly in American suburban architecture, into the early twentieth century.
Revival: "All revivals are romantic" says J. Mordant Cook. What makes a "revival" style is that it is a self-conscious effort to recall, reflect or associate with something that once was.
Vernacular: Usually refers to buildings whose design more strongly reflects cultural and local traditions than architectural ideals. Vernacular houses can reflect and borrow from architectural styles and trends. This can be frequently observed in 18th and early 19th century dwellings. Sometimes the reverse is true. That is an architect draws on vernacular examples for inspiration, design, or details.
Exhibit by Matthew Grubel, 2008