There are two main styles of adzes before the micro-definitions are explained. The hand adze, which is a hand tool with a short handle about the length of the average hammer handle. And the foot adze which has a longer handle similar to an axe handle.
The blade of an adze is built at a right angle to the shaft of the tool. Imagine an axe with the blade itself twisted one turn at 90 degrees.

adze (n.)
also adz, "cutting tool used for dressing timber, resembling an axe but with a curved blade at a right-angle to the handle," 18c. spelling modification of ads, addes, from Middle English adese, adse, from Old English adesa "adze, hatchet," which is of unknown origin. Adze "has been monosyllabic only since the seventeenth century. The word has no cognates, though it resembles the names of the adz and the hammer in many languages". Perhaps somehow related to Old French aisse, Latin ascia "axe" (see axe).1

English:

Boat-Building Scene, ca. 664-634 B.C.E. Limestone, painted, 7 5/8x10 5/8 in. (19.4 x 27 cm).

The adze is believed to have originated in Egypt. The original adzes had a stone blade fastened to a wooden handle by tying's or wraps of twine like material or leather. The adze can be found on ancient hieroglyphs in Egypt and surrounding regions.

Prehistoric Māori adzes from New Zealand, used for wood carving, were made from nephrite, also known as jade. At the same time on Henderson Island, a small coral island in eastern Polynesia lacking any rock other than limestone, natives may have fashioned giant clamshells into adzes.2

American Northwest coast native peoples traditionally used adzes for both functional construction (from bowls to canoes) and art (from masks to totem poles).
Northwest coast adzes take two forms:

• The first form is the Hafted. The hafted form is similar in form to a European adze with the haft constructed from a natural crooked branch which approximately forms a 60% angle. The thin end is used as the handle and the thick end is flattened and notched such that an adze iron can be lashed to it. Modern hafts are sometimes constructed from a sawed blank with a dowel added for strength at the crook.

• The second form is the D-handle. Adze which is basically an adze iron with a directly attached handle. The D-handle therefore provides no mechanical leverage. Northwest coast adzes are often classified by size and iron shape vs. role. As with European adzes, iron shapes include straight, gutter and lipped. Where larger Northwest adzes are similar in size to their European counterparts, the smaller sizes are typically much lighter such that they can be used for the detailed smoothing, shaping and surface texturing required for figure carving. Final surfacing is sometimes performed with a crooked knife.2

Ground stone adzes are still in use by a variety of people in Irian Jaya (Indonesia), Papua New Guinea and some of the smaller Islands of Melanesia and Micronesia. The hardstone is ground on a riverine rock with the help of water until it has got the desired shape. It is then fixed to a natural grown angled wood with resin and plant fibers. The shape and manufacture of these adzes is similar to those found from the Neolithic stone age in Europe. A variety of minerals are used. Their everyday use is on a steady decline, as it is much more convenient to cut fire wood using imported steel axes or machetes. However, certain ceremonial crafts such as making canoes, ceremonial shields, masks, drums, containers or communal houses etc. may require the use of traditional-made stone adzes.2

The traditional way to use an adze is to straddle the timber or log while holding the adze by its long handle and swing the adze up out in front and center to the users body, then swing the adze down to strike the timber or log. There are many ways to use a long handled adze and various stances or positions for the user to take as the work dictates.
Foot adzes are most commonly known as shipbuilder's or carpenter's adzes. They range in size from 00 to 5 being 3 1/4 to 4 3/4 pounds with the cutting edge 3 inches to 4 1/2 inches wide. On the modern, steel adze the cutting edge may be flat for smoothing work to very rounded for hollowing work such as bowls, gutters and canoes. The shoulders or sides of an adze may be curved called a lipped adze, used for notching. The end away from the cutting edge is called the pole and be of different shapes, generally flat or a pin pole.2

Shipwright Herbert Page from Hastings uses a foot adze to trim and shape a large timber.

An adze with a short handle is used for hand work of smaller wooden projects and items. For example the adze with the shorter hammer style handle is used to rough shape the concave of a chair seat.

Hand adze with a hollowing head. Peter Follansbee hollowing a Welsh chair seat. The adze Mr. Follansbee is using is a hand adze.

A carpenters adze is used for smoothing large surfaces and bringing timbers to rough dimension or for smoothing one or all sides of a large timber. This adze is also used for leveling boards for flooring and for hewing large post and beams for building construction.

The blade of a carpenters adze is typically flat and the weight of the carpenters adze is considerable, the heft in this adze is beneficial for having the leverage and swing in order to remove large amounts of material quickly.

Cooper's adze, mounted as exhibit. This adze was in use by Jules Van den Bossche, the last cooper of Willebroek, Belgium.

3 The coopers adze is typically a short handled adze used with one hand. It has a hammer face on the opposite end of the adze for setting the hoops in place over the barrel. This adze is used for barrel making mainly, the 4 chime of the barrel is created with the coopers adze. The chime is beveled at the top and bottom of the barrel where the end of the staves come together.
4 The coopers adze will have a bolt that is inserted through the entire length of the handle. The cutting edge of the coopers adze is typically 2" to 3 1/2".

Short handled gutter adze by Gränsfors Bruk, Sweden

The gutter adze is aptly named for the gutter shape form that it makes during use. The gutter adze can be used for shaping out canoes, chair seats and any other type of work that requires an elongated scooped out form.

Ships Carpenters Adze

Adzes are used to chip away at the surface of the timber, either to reduce its size or to shape it to a particular dimension. Depending on the size of the adze and curvature of the blade, a skilled adzeman could plane the surface of the timber quite smoothly with little trace of the adze blade.
In this 1927 photo these ships carpenters are using adzes on structural timbers for the USS Constitution‘s extensive 1927-1931 restoration. Courtesy U.S. Navy5

• The Adze And The Ungrooved Axe Of The New England Indians