The term overcoat dates from the time when a jacket was called an undercoat, was worn all the time, and an overcoat was a garment that went on over the jacket. Jackets and coats are now generally thought of as exterior items, so the categories can get a little jumbled as to what's an overcoat and what's an undercoat.
The trenchcoat was arguably the most popular overcoat between 1940 and 1960. A long coat meant to go to the knees, trenchcoats were originally designed for soldiers in the front "trenches" in World War I, hence the name. The style has persisted, however, and these coats are still popular with both men and women. They can also be made out of a variety of materials, from the traditional gaberdine to a more edgy leather. The coat has a collar that can be pulled up, and original coats had straps and rings for a soldier to carry equipment.
The duster coat was common in the American West, and it's a fixture of Western films and books. A duster was worn over the clothes to keep trail dust from getting on a rider's outfit. As such, dusters were easily washed, and canvas was a common material. Dusters typically buttoned up the middle, and they had a short cape for dust or rain. The back was usually split so it could be worn while riding a horse.
A number of overcoats are meant to offer protection to the wearer. Leather coats were worn as armour in the Middle Ages, and today they're used to protect motorcycle riders in the event of a fall. Raincoats, which are often made of oil skins or rubberised fabric, keep the wearer dry. Thermal overcoats, made with wool or down, keep the wearer warm.
The frock coat, which is considered fairly old-fashioned by modern standards, is a dress overcoat. Cut like a frock, with two wide lapels, the coat typically reached a little past the knee and earlier versions had no external pockets. These coats were popularly seen in England, as well as on the captains of industry of American business in the early parts of the 20th century.
Another style of overcoat is the Ulster. This coat boasts a relatively small and high collar, with a short cape around the shoulders. The coat went below the knees, and it bore external pockets. It could be made of warm wool for winter wear, or an ulster could be made of oil cloth to be used as a rain slicker.