There are many of these classification systems. Robert Bringhurst, in the currently accepted standard reference on typography, The Elements of Typographic Style, Hartley and Marks, 1992, uses historical markers. He sets up categories based largely on the historical periods of fine art: scribal or Carolingian, Renaissance, Mannerist, Baroque, Rococo, Neoclassical, Romantic, Realist, Geometric Modernism, Expressionist, Elegiac Post-Modernism, and Geometric Post-Modernism. It is, indeed, a fascinating journey through history. Robert presents his case extremely well. He’s an excellent writer and a poet. You will acquire a great deal of useful knowledge by reading his book.
It is a bit over the top, though. Plus, he skips a lot. Bringhurst clearly does not like Victorian letter styles, so he does not mention them. He skips all of the modern variations like Art Nouveau, Art Deco, and the more extreme elements of the early twentieth century. He barely mentions the slab serifs of the late nineteenth century because he thinks they are coarse. In other words, he presents the type he likes in an excellent setting. It is not very useful for us though.
Thomas Phinney, now of Adobe, wrote a nice little historical piece on a site now gone. In it, he uses the more common set of categories currently taught in many design schools: Old Style, Transitional, Modern, Slab Serif (or Egyptian), fat faces, wood type, Art Nouveau, Art Deco, synthesis, and grunge. This covers everything nicely and we will follow that basic lead.
The importance of classification has to do with appropriateness. Bringhurst is still over the top here when he suggests using French type for French products and so on. What we should be using is historical type in context. You cannot pull off a Western “Wanted” poster with anything but Victorian type from the late 1880s. All of the recent Retro looks have specifically used fonts from a historical period placed in a hip, fashionable setting. Within a few months there were Retro fonts specifically designed to match the style. At this point, in graphic design, the fonts appear along with the new fashion.
The fonts we are most comfortable reading are those based on character forms from the fifteenth to the eighteenth centuries. I include most of the font designs developed for early printers (except for the manuscript styles usually called Blackletter). These start with the earliest fonts from the Renaissance that were first converted from scribal forms shortly after Gutenberg’s Bible and continued for a century or so.
These original serif fonts are exemplified by the work of Claude Garamond in Paris in the early to mid-1500s. Robert Slimbach recently released a masterful new interpretation of Claude Garamond's and Robert Granjon's styles. This type of font is the standard to which all other fonts are compared. They are full of smooth sensuous curves. They are light, and open — beautiful, comfortable, and elegant. The stems are vertical. The bowls are nearly circular. The crossbars often rise to the right (but usually only with the e). The axis is mostly humanist until the end of the period in the mid-1700s. The aperture is comfortably open. There is enough contrast to help but not enough to dominate (as we'll see in the Revolutionary fonts next. They are direct descendants of the incredible calligraphic work of the fourteenth and fifteenth centuries.
They had little to do with what I am calling Oldstyle fonts. The first italic font cut was by Griffo, commissioned by Aldus Manutius, in 1499. These early italics seem very condensed to us, with elliptical bowls and very calligraphic stroke endings (as opposed to actual serifs). Oddly, many of these fonts have vertical capital letters. This was the cursive handwriting of the period. They do not make good companions to modern fonts. They were closer to what we now call script.

Through the 1500s and 1600s, these old style letter forms went through gradual changes. Designers began playing with the forms. Sloped capitals were added to the italics. Paired roman and italic fonts appeared. As Europe was caught up in the extravagance and luxury of the Baroque and Rococo, those lavish curves and flourishes made their way into type design as well, as we see in Caslon, among others.
Type design gradually became drawn rather than written. Baroque designers played with letter forms, having stems that varied in slope and bowls that varied in axis in the same font. The entire period was extravagant, but tightly based on classical old styles.
By the end of this period, fonts had appeared with a rigidly vertical axis (usually called a rational axis). You can see it in Baskerville above. This was the time of the Revolution and design was into Retro classical, which was called Neoclassical by the historians. This is the time of Monticello. Franklin was extremely impressed with John Baskerville’s designs in England at the time.
Throughout this period, careful adjustments were tried with axis, aperture, serif style, and so on. However, to our eye in the twenty-first century, all of these fonts are minor variations on a common theme.
Old style fonts are still the normal choice for body copy. Your personal style will determine which you choose. The variations definitely have their own character and leave their feel in the documents that use them. Beyond that, they are all old, traditional letterforms to us. The main point is that all of these fonts, to the contemporary eye, look very similar. More to the point, they all provoke nearly identical reader reactions (unless that reader is quite sophisticated graphically — with a trained eye). It is true there are major differences in the typographer’s eye, but then there are not many of us. Functionally, these all can be used in the same places, for the same clients. The only differences are ones of taste & personal style.

These are type styles of the late 1700s and early 1800s, although their influence remains. To call them Modern, as most of the schools do, is silly. They are 200 years old. To call them Romantic (as Bringhurst does) is equally strange for they are cold fish. They are the natural expression of the radical, revolutionary intellectualism of the period. They are built with hard, tightly structured letterforms which push out the emotional, warm, comfortable type of the Old Style fonts, replacing it with spiky, carved, structured forms. Serifs lose all bracketing, becoming thin, horizontal lines. The aperture is shut down quite a bit. The axis is rigidly vertical and accented with often extreme stroke modulation.
These fonts can be very beautiful, but never comfortable. Baskerville led into this but it is still a very conservative, old style font when compared to these. Most touches of humanity are cleaned out of the revolutionary styles. The best you can do is think of a severe elegance — a cold formality. Actually, Didot above is not as extreme as some of the versions of Bodoni that have been revived. Actually, from these rare book samples from the 18th century, you can see that all of our fonts have been cleaned and lightened substantially.

In the mid-1800s, a type design movement began making type for the workers, the common man, the noneducated. They were never really popular with designers, but they have had a lot of influence. They are based on the revolutionary fonts but distinguished by an even stroke weight with very little modulation. The aperture is nearly closed. Serifs are close to being unbracketed slabs with the same stroke thickness as the rest of the letterform. There are no small caps, old style figures, ligatures, or any of the other graceful tools of the now traditional typography.
One of these fonts, Century Schoolbook, is the font many of us used when we learned to read in the first few grades of school. It may be the most elegant of the bunch. In general, heavy, clunky, and old-fashioned are the terms associated with fonts like these. Typical fonts would be Bookman, Cheltenham, or Clarendon. They are sometimes called Victorian, but Victorian styles are most commonly in the Display or Decorative classifications as used today.
The swirling curves of Art Nouveau were also used to produce type. It seems like a rebellion against the realists, as the entire movement was a rejection of contemporary morality and tradition.

These are the first fonts with little tie to traditional letterforms. They were never really popular, but certain cultures can take them on for a time. They were the absolute standard for the Spanish culture in New Mexico in the early 1980s, for example. An Art Nouveau font would be Arnold Böcklin, but Raphael is usually liked by the same people though it is not really from the period of time.
Fonts like these need to be used very carefully. From a design point of view, they are very interesting. However, they are usually quite hard to read. The larger problem, though, is their ties to a cultural movement known for its depravity. They are used a lot by writers and designers in the occult. You need to be aware of these issues.

These fonts are an outgrowth of the modernist movement of the early twentieth century. Here letter forms are constructed geometrically, with purely circular bowls, no modulation, slab serifs, closed aperture, and so on. Intellectually, they could almost be considered the scientific extension of the socialist expression found in realism.
These are often called Egyptian fonts simply because many of them were designed around the time of the great Egyptian discoveries in the same period as Art Nouveau. Typical fonts of this type are Memphis, Rockwell, and City. The readability is usually very low. However, the serifs make these fonts a much better choice for readability than geometric sans fonts like Avant Garde, Century Gothic, and even Futura.
Many recent serif faces play with attributes of any and all historical styles. Often they experiment with distinctive serif stylings, sharp angular features, fanciful modulations. However, these more playful aspects are often very restrained and elegant. They take pieces from all over and show a wide variety from Times New Roman to Palatino to Veljovic. Often, like in Usherwood, the x-heights are very large — strictly a fashion statement from the 1980s.
There are not nearly as many options in sans serif type. I am only going to give you four general types. I frankly invented these categories to help you make sense of what you run across in your search to build your own font library.

These are largely a product of radical modernism, the Bauhaus in Germany, and the Art Deco movement of the 1920s and 1930s. The letterforms seem to be absolute geometric constructs, but they often have many subtle adjustments beyond that to make them more readable (but just barely).
The real problem with geometric fonts is the readability issue. Because all of the bowls are perfectly round and the aperture is usually almost closed, there is little visual difference between an e or an o. More than that, an ol looks a lot like a d, an rn can look identical to an m, and even a cl can seem to be a d. They can work fairly well in headlines, but using them for body copy is usually a serious mistake. Typical fonts are: Futura, Kabel, Avant Garde, and Bauhaus.
These are what I call the normal fonts like Helvetica and Univers. (Arial/Geneva are the Microsoft/Apple versions of Helvetica.) Most of them have many subtle curve adjustments, but the stroke is virtually unmodulated. The aperture is closed up tight in most of them. In general, even the best of them "feel clunky", for lack of a better word.
They are difficult to read and cause what I call "bureaucratic" reactions in the reader—simply because so many bureaucracies require their use. Helvetica can be used well, but you have to be very careful.
This is what I am calling those fonts that have a style that seems relatively warm and friendly, even though there is little or no modulation of the stroke. As a result there is no axis that could be called humanist. Many of these fonts make relatively good body copy in short bursts. They all have a distinctively warm feel — relative to other sans serif faces. Common faces in this genre would be Gill Sans, Frutiger, Corinthian, Skia, and Trebuchet — among many others. Myriad has become the Adobe default sans. They are very popular for good reason. Dell used Gill Sans to distance itself from the Helvetica of the typical business PC competitors, and to seem more friendly, warm and accessible.

These fonts are actually neither fish nor fowl. Instead of serifs they tend to have slight flares, They have a modulated stroke and a humanist axis. They are the most elegant of the sans serifs. Most commonly available would be Optima, Poppl Laudatio, and Zapf Humanist.
Humanist sans serifs are growing in popularity. They are quite readable. They may well become the fashion for body copy in the near future. Stylized and Humanist sans serif typefaces are very clean, neat, and unobtrusive. They've been one of the focuses of my designs in the later part of of the 1st decade of the new millennium. They have been customer favorites, especially Brinar. You should give them a try.
What about all the type that is outside the classifications we just covered? First of all, proportionally there isn’t that much of it. Most of it is either serif or sans serif anyway. However, there is huge variety, in every artistic style, for every historical period. Many are so rigidly categorized that they can hardly be used for anything else.
Decorative is the term for this miscellaneous grab bag. Decorative type is defined as typefaces that are so highly stylized that they cannot be read in body copy sizes. You need to be very careful in the use of these fonts. Legibility is the obvious problem.
These are a few descriptive terms I'll be using to talk about some of the differences between the categories.
There are more, but this will be enough for our purposes. As you can see, type gets very technical. The differences will seem insignificant to you now, as you start. But they are really very important.
Aperture, for example, tends to control the friendliness and readability. The axis changes from humanist to mechanical vertical strongly influence our reaction to the warmness or coolness of a font. Strong contrasts in modulation can cause severe printing problems in fonts like Bodoni.