To my mind, Classical, Neo-Classical, Baroque, Rococo, and Gothic refer primarily to a set of related architectural styles. These words are also used to refer to painting, music, and literature; but to me they refer to architecture.
“Classical” refers to an extremely precise (and tiny) set of architectural elements that can make up the exterior of a grand building. There are exactly three different kinds of columns you can use, in order of increasing fanciness: Doric, Ionic, or Corinthian. If you’re operating on a shoe-string budget, there is technically an even less fancy kind of column called Etruscan (or Tuscan). And if you you’re feeling enthusiastic, you can create a “Composite” column by combining elements from the other types.
These columns are used to hold up very short stone beams which in turn hold up the roof. (The construction of the roof is an engineering issue and not considered relevant.)
The short stone beams between the columns are mostly simple extrusions which can be embellished in a very particular way with carved vertical stripes and plain rectangles, which can in turn be decorated with carvings or paintings.
The walls behind the columns are generally plain and smooth, though they may be embellished with very simple geometric patterns. The base of the building is constructed of more roughly-textured stone. There are no windows; only doors. The interior is generally a single large room.
The building must be set on a flat or nearly-flat site.
The term “classical” has almost no meaning when referring to interior architecture. If the interior of a building is “classical”, this means the interior is designed to imitate the exterior.
Classical (Greek) architecture, built from stone, is believed to have evolved from earlier Greek architecture (none of which is extant) made from wood. They adapted the techniques they used when building with wood and retained what had previously been structural requirements as stylistic embellishments. For instance, the decorative vertical stripes on the beams were meant to visually emulate the ends of what had once been structurally necessary rafter ties.
The figurative decorative elements of the Ionic and Corinthian columns (the scroll and the leaf motifs) are believed to have been culturally and iconographically significant to the ancient Greeks. They simply decorated the tops of their columns with things they thought were significant. Vitruvius claims that the top of a Corinthian column was meant to resemble a basket filled with mementos left on the grave of young girl. According to the story, an acanthus plant sprouted underneath the basket and intertwined with it.
Technically, classical architecture was brightly painted in ancient times, but the paint has peeled off and we’ve been left with plain white stone.
Classical architecture is characterized by extremely simple geometric shapes such as squares, rectangles, and circles. Almost the entirety of classical architecture consists on variations of these two forms:
(The ancient Romans and Greeks built structures with more complicated floorplans of course, but the term "Classical architecture" refers mainly to religious buildings.)
A temple would generally only be built somewhere where the peak of a mountain or tall hill could be seen while looking straight out the entrance and straight out the back. This had some sort of religious or symbolic importance to the ancient Greeks and Romans.
Neo-Classical refers to the attempt to recreate the grandeur of ancient Greek and Roman architecture during the Renaissance (and pre-Renaissance) period. It is practically identical to Classical architecture except that building techniques had evolved (or devolved) and more complex shapes were permissible. There was also considerable leeway for more complicated floorplans, irregular site elevations, and windows; but symmetry was still key. Wherever possible, the shapes and visual language of classical architecture were adapted for new functions: for example, a window frame could be designed to resemble a stripped-down version of the front of a Greek temple.
Baroque and Rococo
Baroque and Rococo are generally considered subsets of Neo-Classical architecture, though in my opinion they are more like its evolutionary descendants. To some people, Baroque and Rococo might be indistinguishable due to their highly decorative nature (and indeed there is some overlap), but Baroque can be characterized as more “masculine and logical” while Rococo is more “feminine and whimsical”.
Baroque is like the legitimate heir of Neo-Classicalism: the older brother with responsibilities that he intends to fulfill; while Rococco is a like an overenthusiastic younger brother who knows the rules but treats them more like guidelines.
Baroque architects ornamented their buildings by piling elements of Neo-Classical architecture on top of each other to the Nth degree, combining them in novel ways, cramming in as much decoration as possible. If they wanted a more complex shape, they built it up out of straight lines, rectangles and circles.
In contrast, Rococo architects felt much less confined by straight lines and simple geometric shapes. They basically did whatever they wanted and applied Neo-Classical design language to it as an afterthought. They would also combine elements of Neo-Classical design together in unexpected ways, like slapping a statue onto a column.
To my mind, the quintessential example of rococo is the spiral columns of Bernini’s Baldacchino.
Gothic architecture marks the point at which Neo-Classical design language began to be inadequate for the engineering and structural requirements of grand architecture. (Non-intuitively, the Gothic period was contemporary to the Baroque and Rococo era... and also pre-dated it.) The Cathedrals were getting too tall, too narrow, and too airy. It became impractical trying to force-fit neoclassical design elements into buildings whose designs were increasing dictated by their engineering requirements and whose proportions were no longer composed of plain, sturdy rectangles.
Classical columns hung around in the interiors, but the exteriors of the buildings were increasingly characterized by massive stained glass windows and flying buttresses, something totally alien to Classical design. Novel engineering problems allowed architects and engineers to come up with novel solutions and develop a new design language built of spikes and diagonals that was still highly decorative but less dependent on tradition.
In my opinion, rosette windows in particular allowed the designers a venue for combining lofty aesthetic and brute engineering considerations together in a way that had never been seen before and may never be seen again.