What should one consider when choosing a style of tripod head?

What should one consider when choosing a style of tripod head? There are ball heads, 3-way pan/tilt heads, geared heads, even gimbal heads.

I'm not asking so much about brands, but styles... how should one consider which type of tripod head will be best suited for their style of photography?

Answer

Your last sentence pretty much describes what you need to consider with tripod heads: what and how do you like to shoot.

As geared and gimbal heads are specialized, it's really between the pan-tilt and ball heads. Many start with pan/tilts due to relatively low costs, but will find themselves switching to the ball head for ease of use, and sturdier head (IMO ball heads offer a more sturdier mount than pan/tilts at the same price point--ie to get a sturdier pan/tilt you have to pay for a more heavy duty head).

3-way Pan/Tilt Head
With 3-way heads, your camera moves on 3 distinct axes (axissesis?). If you need precise movement on each axis for your needs (panning for sports for example), then a 3-way makes a lot of sense. It's downside is the need to adjust three knobs to set your camera position. Many start out with a Pan/Tilt head, because of the lower costs.

Ball Head
A ball head gives you less precise control of each axis, because your camera is mounted on a ball and socket joint, typically with two controls, one to lock the mount, and the other to adjust tension. You have less control, but are given more degrees of freedom to move your camera around, allowing you to more quickly adjust your camera with fewer controls to manipulate. Some ball heads allow for independently panning around the horizontal axes without manipulating the actual socket, but this is a feature, not a standard. This is probably the most general purpose type of tripod head, and many who start with budget pan/tilt heads move to ball heads for greater flexibility and sturdier mounting.

Geared Heads
A geared head is a type of head that uses a crank-shaft + gears to rotate the head. Typically found on 3-way heads, these heads allow you to rotate the head using a crank, rather than applying force to the camera body itself, resulting in smoother movement, and a finer grain of control over moving the camera body. Architecture, and any other application requiring a fine control over placing the camera make for good uses for geared heads.

Gimbal Head
Gimbal heads are specialized heads typically reserved for long telephoto lenses (400mm+). It gets it's name from a gimbal, a type of pivot support that allows movement on a single axis. Gimbal heads are designed to allow you to rotate heavy lenses about it's center of gravity, which results in easier movement. These are used in any application that uses huge telephotos, bird-in-flight, other wildlife, sports, etc.