The following timeline gives you an outline of the developments in the various horse species and genus, from the first recognised equid to the modern Equus.

First Equids - The Dawn Horse (Eocene Period)

Eohippus \ Hyracotherium

The first recognised member of the horse family (equid) was Hyracotherium, a small forest animal of the early Eocene.  Initially this little fossil animal was called Eohippus "Dawn Horse".  

The Eocene period ran from approx 54-34 million years ago. This little animal (10-20" at the shoulder) looked more "doggish" than like a modern horse.  It had an arched back, short neck, short snout, short legs,4 toes on each front foot, 3 on hind feet and a long tail. It browsed on fruit and fairly soft foliage, and probably scampered from thicket to thicket like a modern small deer, only stupider, slower, and not as agile. 

Orohippus

In the early-middle Eocene, there was a smooth, gradual transition from Hyracotherium to a close relative, Orohippus. Overall, Orohippus looked much like Hyracotherium, however, the vestiges of the 1st and 2nd toes vanished. The most significant change was in the teeth, indicating Orohippus was eating tougher plant material.

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Medium-Sized Browsing Horses (Late Eocene & Oligocene)

As we move toward the Oligocene, horses start to change. The climate of North America was becoming drier, and grasses were just evolving. The vast forests were starting to shrink. The late Eocene horses responded by developing tougher teeth and becoming a bit larger and leggier (for better speed out in the open).

Mesohippus

The species Mesohippus celer appears suddenly in the late Eocene, approx 40 My (such sudden speciations can occur when a population encounters new selective forces and/or becomes isolated from the parent species. These speciations are "sudden" only in geological terms, of course, where a few million years is "sudden".) This animal was slightly larger than Epihippus, 24" at the shoulder. It didn't look as doggish, either. The back was less arched, the legs a bit longer, the neck a bit longer, and the snout and face distinctively longer. Mesohippus had three toes on its hind feet and on its front feet -- the 4th front toe was reduced to a vestigial nubbin. 

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Miohippus

Soon after Mesohippus celer and its very close relative Mesohippus westoni appeared, a similar animal called Miohippus assiniboiensis arose (approx. 36 My). This transition also occurred suddenly, but luckily a few transitional fossils have been found that link the two genera. A typical Miohippus was distinctly larger than a typical Mesohippus, with a slightly longer skull. The facial fossa was deeper and more expanded. In addition, the ankle joint had changed subtly.

It was once thought that Mesohippus "transformed" gradually into Miohippus via anagenetic evolution, so that only Miohippus continued. Recent evidence shows that instead, Miohippus speciated (split off) from early Mesohippus via cladogenetic evolution, and then Miohippus and Mesohippus overlapped for some 4 million years. For instance, in one place in modern Wyoming there were three species of late Mesohippus coexisting with two species of Miohippus. (Prothero & Shubin, 1989)

The Miohippus Radiation (Early Miocene 24)

Mesohippus finally died out in the mid-Oligocene. Miohippus continued for a while as it was, and then, in early Miocene (24 My) began to speciate fairly rapidly. The horse family began to split into at least 2 main lines of evolution and one small side branch:

1. 3-toed browsers called "anchitheres". They were very successful, spread into the Old World, and thrived for tens of millions of years. They retained the small, simple teeth of Miohippus. Genera include Anchitherium and the large Hypohippus and Megahippus.
2. A line of small "pygmy horses", e.g. Archeohippus. These horses did not survive long.
3. A line that underwent a transformation from browsing to grazing, taking advantage of the new grasses. Large grasslands were just beginning to appear, thus creating a new ecological "opportunity" for grazers. Grass is difficult to chew and wears down teeth rapidly (due to the silica in the leaves) and thus a grass-eater needs tough teeth with ridges of some sort. Open-country grass eaters, in addition, often benefit from being swift runners with long legs. The evolution of this line of horses is described below.

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Horses Move Onto the Plains: Spring-Foot & High-Crowned Teeth (Miocene)

As this third line of Miocene horses began to specialize in eating grasses, several changes occurred. First, the teeth changed to be better suited for chewing harsh, abrasive grass. Small crests on the teeth enlarged and connected together in a series of ridges for grinding. There was a gradual increase in the height of the tooth crowns, so that the teeth could grow out of the gum continuously as the tops were worn down ("hypsodont" teeth). And, in addition, the tooth crowns became harder due to the development of a cement layer on the teeth.

Second, these horses started to become specialized runners. There was a simultaneous increase in body size, leg length, and length of the face. The bones of the legs began to fuse together, and the leg bones and musculature became specialized for efficient forward-and-back strides, with flexible leg rotation being eliminated. Most significantly, the horses began to stand permanently on tiptoe (another adaptation for speed); instead of walking on doglike pads, their weight was supported by springy ligaments that ran under the fetlock to the big central toe. All these changes occurred rapidly, and we are lucky to have a fairly good fossil record during this time. This was one of the most interesting times in horse evolution. 

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Parahippus

Approx 23 million years ago Parahippus was a little larger than Miohippus, with about the same size brain and same body form. Parahippus was still three-toed, and was just beginning to develop the springy ligaments under the foot. Parahippus showed gradual and fluctuating changes in its teeth. Parahippus evolved rapidly and was quickly transformed into a fully spring- footed, hypsodont grazing horse called Merychippus gunteri. This burst of evolution took place about 18-17 My. Later fossils of Parahippus (e.g. the species Parahippus leonensis) are so similar to early Merychippus that it's hard to decide where to draw the line between the genera.

Merychippus 

A typical Merychippus was about 10 hands (40") tall, the tallest equine yet. The muzzle became elongated, the jaw became deeper, and the eye moved farther back, to accommodate the large tooth roots. The brain was notably larger, making Merychippus a smarter and more agile equine than the earlier horses. Overall, Merychippus was distinctly recognizable as a horse, and had a "horsey" head.

Merychippus was still 3-toed, but was fully spring-footed. This animal stood permanently on tiptoe, supported and propelled by strong, springy ligaments that ran under the fetlock. The side toes were still complete, but began to be of varying sizes; some Merychippus species had full-size side toes, while others developed small side toes that only touched the ground during running. The central toe developed a large, convex, "horsey" hoof, and the legs became longer.  All these changes made Merychippus' legs specialized for just one function: rapid running over hard ground.

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The Merychippine Radiation (Miocene, 15 Million years ago)

Merychippus gave rise to at least 19 new grazing horse species in three major groups. This explosive burst of horse evolution is often called the "merychippine radiation". 

About 10 Million years ago, the horse family reached an apex of diversity (of species and of genera) and sheer numbers which it has never equalled since. The Old and New Worlds both seemed overrun with a wide variety of hipparions, protohippines, and "true equines", large and small, forest browsers and plains grazers. Throughout the evolution of all these related merychippine descendents, the facial fossae got deeper and more elaborate. With so many equine species overlapping at once, these facial fossae may have housed species-specific glands of some sort, similar to the scent- marking glands of modern antelopes and deer.

One-Toed Horses (Late Miocene, Pliocene & Pleistocene)

The late merychippine species of this line, such as M. carrizoensis, were large horses with small side toes. They gave rise to at least 2 separate groups of horses that independently lost their side toes. This occurred as side ligaments developed around the fetlock to help stabilize the central toe during running. These one-toed horses include: Pliohippus, Astrohippus and Dinohippus

Dinohippus

Finally, a one-toed horse called Dinohippus (recently discovered) arose about 12 Million years ago. The exact ancestor of Dinohippus is not yet known. They look just like Equus in foot, teeth, and skull. Throughout the end of the Pliocene, Dinohippus showed a gradual decrease in the facial fossae, straightening of the teeth, and other gradual changes, as Dinohippus smoothly graded into Equus. (Hulbert, 1989)

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Equus

Finally we arrive at Equus (4 Million years ago), the genus of all modern equines. The first Equus were approx. 13.2 hands tall (pony size), with a classic "horsey" body -- rigid spine, long neck, long legs, fused leg bones with no rotation, long nose, flexible muzzle, deep jaw. The brain was a bit larger than in early Dinohippus. Like Dinohippus, Equus was (and is) one-toed, with side ligaments that prevent twisting of the hoof, and has high-crowned, straight grazing teeth with strong crests lined with cement.

The earliest known Equus species were a set of three "simple Equus" species collectively known as the Equus simplicidens group. They had zebra-like bodies (relatively stocky with a straight shoulder and thick neck), and short, narrow, donkey-like skulls. They probably had stiff, upright manes, ropy tails, medium-sized ears, striped legs, and at least some striping on the back (all traits shared by modern equines). They quickly diversified into at least 12 new species in 4 different groups, in a burst of evolution. All these Equus species coexisted with other one-toed horses (such as Astrohippus) and with various successful hipparions and protohippines, which had been merrily evolving on their own paths.

Modern Equines

The three-toed horses gradually died out and most of the one-toed horses in North America also died out, as the Ice Ages started. However, one-toed Equus was very successful. Until about 1 million years ago, there were Equus species all over Africa, Asia, Europe, North America, and South America, in enormous migrating herds.  In the late Pleistocene there was a set of devastating extinctions that killed off most of the large mammals in North and South America. All the horses of North and South America died out, for the first time in tens of millions of years, there were no equids in the Americas.

The only members of Equus -- and of the entire family Equidae -- that survived to historic times were:

Equus burchelli: the Plains zebra of Africa, including "Grant's zebra", "Burchell's zebra", "Chapman's zebra", the half-striped Quagga, and other subspecies

Equus grevyi: Grevy's zebra, the most horse-like zebra 

Equus caballus, the true horse, which once had several subspecies

Equus hemionus: the desert-adapted onagers of Asia & the Mideast

Equus asinus: the true asses & donkeys of northern Africa