Digital camera modes

Most digital cameras support the ability to choose among a number of configurations, or "modes", for use in various situations. Professional DSLR cameras provide several manual modes; consumer point-and-shoot cameras emphasize automatic modes; amateur prosumer cameras often have a wide variety of both manual and automatic modes.
Manual-enabled modes.
Manual-enabled modes give the photographer control over the various parameters of exposure. There are three exposure parameters – aperture, time (shutter speed), and sensitivity (ISO), and in different modes, these are each set automatically or manually; this gives 23 = 8 possible modes. For a given exposure, this is an underdetermined system, as there are three inputs but only one output. Accordingly, there are many combinations that result in the same exposure – for example, decreasing the aperture by one stop but increasing the exposure time or sensitivity to compensate, and there are various possible algorithms to automatically choose between these.
Most often, ISO is considered separately, being either set manually or set to Auto ISO, and then only aperture and shutter speed need be determined – either determines the other.
The four main modes, sometimes abbreviated "PASM", are:
Together with setting ISO manually or automatically, this (PASM) yields the 4×2 = 8 possible combinations of manual/auto.
Exposure is further controlled in each of the above modes with an independent setting for:
Less commonly seen modes include:
In cases where there is camera discretion (e.g., Auto ISO), different cameras allow different configurations of how decisions are made. For example, , Nikon cameras allow one to set the maximum and minimum ISO sensitivities, and slowest shutter speed that will be used in automatic modes, while Canon cameras will select within the fixed range of ISO 400–ISO 800 in Auto ISO mode. In Nikon cameras, the Auto ISO mode first adjusts the shutter speed, keeping ISO at its minimum desired value, then, when shutter speed reaches the user-defined limit, the ISO is increased, up to the maximum value.
All of the above functions are independent of lens focus and stabilizing methods.
Automatic modes.
In automatic modes the camera determines all aspects of exposure, choosing exposure parameters according to the application within the constraints of correct exposure, including exposure, aperture, focusing, light metering, white balance, and equivalent sensitivity. For example, in portrait mode, the camera would use a wider aperture to render the background out of focus and would seek out and focus on a human face rather than other image content. In the same light conditions, a smaller aperture would be used for a landscape, and recognition of faces would not be enabled for focusing.
Some cameras have tens of modes. Many cameras do not document exactly what their many modes do; for full mastery of the camera, one must experiment with them.
In general:
Secondary modes.
Aside from the main modes which control exposure, there are usually other, secondary settings common to digital cameras.
Drive mode.
"Single" mode will take a single picture each time the shutter button is depressed. "Continuous" or "burst" mode will take multiple photographs in quick succession as long as the shutter button is held down, and is typically used to capture fast-moving subjects such as in sports photography.
Other drive modes commonly available include self-timer, which takes a picture several seconds after the shutter button is pressed; automatic exposure bracketing, used to take multiple photos each with different exposure settings; and remote shutter mode for use with a remote control accessory to trigger the camera at a distance. Some cameras also have an intervalometer mode for time-lapse photography, possibly with the option to automatically create a video from the pictures taken.
Autofocus.
Autofocus (AF) can either activate until a lock is found (single, AF-S) or be continuously active (continuous, AF-C, servo). Single-mode is especially favored for stationary subjects, when focus, once found, should stay fixed, while the continuous mode is used for moving subjects. Some AF systems also include anticipation of the position of moving subjects – Canon calls this "AI servo" (for artificial intelligence) – or can automatically switch between single and continuous depending on whether the subject is moving – Canon calls this "AI focus".
A separate but often related distinction is between and release priority – whether the camera will take a picture when the subject is out of focus or not. In focus priority, the camera will only take a picture when the subject is in focus (as detected by the AF system), while in release priority, the camera will take a picture whenever the shutter is pressed. Release priority is particularly used of fast-moving subjects, which may not be perfectly in focus, or by experienced photographers, who wish to override the camera's judgment of whether the scene is in focus.
These are usually combined: for stationary subjects, AF is set to single (lock when found) and release is set to focus priority, while for moving subjects, AF is set to continuous and release is set to release priority. Manual focus is generally in release priority – AF is neither detected nor set.
Note that these "priority" modes should not be confused with the same word in exposure modes.
Focus priority can also be used for the trap focus trick: to take a picture only when a subject hits a focus point, by using AF to detect focus but not set it.
Back button focus.
In most digital cameras, by default autofocus is only activated (AF-on) when the shutter button is pressed halfway down, which helps to preserve battery life. However, some photographers find that having AF-on and the shutter release on the same button makes it harder to establish the correct focus point, or hold it once the desired point is established. By using a separate shutter and focus button you can lock focus at the same time as shooting without changing exposure, allowing photographers to focus on the subject through the viewfinder. 
One technique to make this easier to control is called "back button focus": separating focus from the shutter control button by reassigning the AF-on function to a button on the back of the camera body. Some models of a camera may have a dedicated AF-ON button, and menu options to disable the focus via shutter button half-press. When photographing at golden hour and fighting off sun flare, the back button focus can become an essential tool to use. 
Back button focus allows you to benefit from the three main possible focusing modes: manual focus, single focus, and continuous focus. It is a technique favorited by many wildlife photographers due to the unpredictability of animal subjects.
Flash.
Flash modes allow the user to choose between common settings such as: "Fill flash", to always use flash; "Auto flash", which will use flash in low-lit areas; "Red-eye reduction", which may flash once before the actual photo in order to shrink the subject's pupils and reduce red-eye; or, "Flash off", which will never use flash.
Flash can have its own exposure compensation–how brightly it flashes–which allows one to independently adjust the exposure of the foreground (lit by flash) and background (out of flash range).