Artillery
Covering fire: used to protect troops when they are within range of enemy small arms.
Defensive fire: delivered by supporting units to assist and protect a unit engaged in a defensive action.
Final Protective Fire: an immediately available prearranged barrier of fire designed to impede enemy movement across defensive lines or areas.
Harassing fire: a random number of shells are fired at random intervals, without any pattern to it that the enemy can predict. This process is designed to hinder enemy forces' movement, and, by the constantly imposed stress, threat of losses and inability of enemy forces to relax or sleep, lowers their morale.
Interdiction fire: placed on an area or point to prevent the enemy from using the area or point.
Preparation fire: delivered before an attack to weaken the enemy position.
These purposes have existed for most of the 20th century, although their definitions have evolved and will continue to do so, lack of suppression in counterbattery is an omission. Broadly they can be defined as either:

Deep supporting fire: directed at objectives not in the immediate vicinity of own force, for neutralizing or destroying enemy reserves and weapons, and interfering with enemy command, supply, communications and observation; or
Close supporting fire: placed on enemy troops, weapons or positions which, because of their proximity present the most immediate and serious threat to the supported unit.

Two other NATO terms also need definition:

Neutralization fire: delivered to render a target temporarily ineffective or unusable; and
Suppression fire: that degrades the performance of a target below the level needed to fulfill its mission. Suppression is usually only effective for the duration of the fire.
The tactical purposes also include various "mission verbs", a rapidly expanding subject with the modern concept of "effects based operations".
Targeting is the process of selecting target and matching the appropriate response to them taking account of operational requirements and capabilities. It requires consideration of the type of fire support required and the extent of coordination with the supported arm. It involves decisions about:

what effects are required, for example, neutralization or suppression;
the proximity of and risks to own troops or non-combatants;
what types of munitions, including their fuzing, are to be used and in what quantities;
when the targets should be attacked and possibly for how long;
what methods should be used, for example, converged or distributed, whether adjustment is permissible or surprise essential, the need for special procedures such as precision or danger close
how many fire units are needed and which ones they should be from those that are available (in range, with the required munitions type and quantity, not allotted to another target, have the most suitable line of fire if there is a risk to own troops or non-combatants);
The targeting process is the key aspect of tactical fire control. Depending on the circumstances and national procedures it may all be undertaken in one place or may be distributed. In armies practicing control from the front, most of the process may be undertaken by a forward observer or other target acquirer. This is particularly the case for a smaller target requiring only a few fire units. The extent to which the process is formal or informal and makes use of computer based systems, documented norms or experience and judgement also varies widely armies and other circumstances.
Surprise may be essential or irrelevant. It depends on what effects are required and whether or not the target is likely to move or quickly improve its protective posture. During World War II UK researchers concluded that for impact fuzed munitions the relative risk were as follows:

men standing – 1
men lying – 1/3
men firing from trenches – 1/15–1/50
men crouching in trenches – 1/25–1/100
Airburst munitions significantly increase the relative risk for lying men, etc. Historically most casualties occur in the first 10–15 seconds of fire, i.e. the time needed to react and improve protective posture, however, this is less relevant if airburst is used.
There are several ways of making best use of this brief window of maximum vulnerability:

ordering the guns to fire together, either by executive order or by a "fire at" time. The disadvantage is that if the fire is concentrated from many dispersed fire units then there will be different times of flight and the first rounds will be spread in time. To some extent a large concentration offsets the problem because it may mean that only one round is required from each gun and most of these could arrive in the 15 second window.
burst fire, a rate of fire to deliver three rounds from each gun within 10 or 15 seconds, this reduces the number of guns and hence fire units needed, which means they may be less dispersed and have less variation in their times of flight. Smaller caliber guns, such as 105 mm, have always been able to deliver three rounds in 15 seconds, larger calibers firing fixed rounds could also do it but it was not until the 1970s that a multi-charge 155 mm howitzer, FH-70 first gained the capability.
multiple round simultaneous impact (MRSI), where a single weapon or multiple individual weapons fire multiple rounds at differing trajectories so that all rounds arrive on target at the same time.
time on target, fire units fire at the time less their time of flight, this works well with prearranged scheduled fire but is less satisfactory for opportunity targets because it means delaying the delivery of fire by selecting a 'safe' time that all or most fire units can achieve. It can be used with both the previous two methods.

Counter-battery fire: Modern counter-battery fire developed in World War I, with the objective of defeating the enemy's artillery. Typically such fire was used to suppress enemy batteries when they were or were about to interfere with the activities of friendly forces (such as to prevent enemy defensive artillery fire against an impending attack) or to systematically destroy enemy guns. In World War I the latter required air observation. The first indirect counter-battery fire was in May 1900 by an observer in a balloon.
Enemy artillery can be detected in two ways, either by direct observation of the guns from the air or by ground observers (including specialist reconnaissance), or from their firing signatures. This includes radars tracking the shells in flight to determine their place of origin, sound ranging detecting guns firing and resecting their position from pairs of microphones or cross-observation of gun flashes using observation by human observers or opto-electronic devices, although the widespread adoption of 'flashless' propellant limited the effectiveness of the latter.
Once hostile batteries have been detected they may be engaged immediately by friendly artillery or later at an optimum time, depending on the tactical situation and the counter-battery policy. Air strike is another option. In some situations the task is to locate all active enemy batteries for attack using a counter-battery fire at the appropriate moment in accordance with a plan developed by artillery intelligence staff. In other situations counter-battery fire may occur whenever a battery is located with sufficient accuracy.
Modern counter-battery target acquisition uses unmanned aircraft, counter-battery radar, ground reconnaissance and sound-ranging. Counter-battery fire may be adjusted by some of the systems, for example the operator of an unmanned aircraft can 'follow' a battery if it moves. Defensive measures by batteries include frequently changing position or constructing defensive earthworks, the tunnels used by North Korea being an extreme example. Counter-measures include air defence against aircraft and attacking counter-battery radars physically and electronically.

Field artillery team: 'Field Artillery Team' is a US term and the following description and terminology applies to the US, other armies are broadly similar but differ in significant details. Modern field artillery (post–World War I) has three distinct parts: the Forward Observer (FO), the Fire Direction Center (FDC)  and the actual guns themselves. The forward observer observes the target using tools such as binoculars, laser rangefinders, designators and call back fire missions on his radio, or relays the data through a portable computer via an encrypted digital radio connection protected from jamming by computerized frequency hopping. A lesser known part of the team is the FAS or Field Artillery Survey team which sets up the "Gun Line" for the cannons. Today most artillery battalions use an "Aiming Circle" which allows for faster setup and more mobility. FAS teams are still used for checks and balances purposes and if a gun battery has issues with the "Aiming Circle" a FAS team will do it for them.
The FO can communicate directly with the battery FDC, of which there is one per each battery of 4–8 guns. Otherwise the several FOs communicate with a higher FDC such as at a Battalion level, and the higher FDC prioritizes the targets and allocates fires to individual batteries as needed to engage the targets that are spotted by the FOs or to perform preplanned fires.
The Battery FDC computes firing data—ammunition to be used, powder charge, fuse settings, the direction to the target, and the quadrant elevation to be fired at to reach the target, what gun will fire any rounds needed for adjusting on the target, and the number of rounds to be fired on the target by each gun once the target has been accurately located—to the guns. Traditionally this data is relayed via radio or wire communications as a warning order to the guns, followed by orders specifying the type of ammunition and fuse setting, direction, and the elevation needed to reach the target, and the method of adjustment or orders for fire for effect (FFE). However, in more advanced artillery units, this data is relayed through a digital radio link.