Artillery
Howitzers can fire at maximum elevations at least close to 45°; elevations up to about 70° are normal for modern howitzers. Howitzers also have a choice of charges, meaning that the same elevation angle of fire will achieve a different range depending on the charge used. They have rifled bores, lower muzzle velocities and shorter barrels than equivalent guns. All this means they can deliver fire with a steep angle of descent. Because of their multi-charge capability, their ammunition is mostly separate loading (the projectile and propellant are loaded separately).
That leaves six combinations of the three criteria, some of which have been termed gun howitzers. A term first used in the 1930s when howitzers with a relatively high maximum muzzle velocities were introduced, it never became widely accepted, most armies electing to widen the definition of "gun" or "howitzer". By the 1960s, most equipment had maximum elevations up to about 70°, were multi-charge, had quite high maximum muzzle velocities and relatively long barrels.
Mortars are simpler. The modern mortar originated in World War I and there were several patterns. After that war, most mortars settled on the Stokes pattern, characterized by a short barrel, smooth bore, low muzzle velocity, elevation angle of firing generally greater than 45°, and a very simple and light mounting using a "baseplate" on the ground. The projectile with its integral propelling charge was dropped down the barrel from the muzzle to hit a fixed firing pin. Since that time, a few mortars have become rifled and adopted breech loading.
There are other recognized typifying characteristics for artillery. One such characteristic is the type of obturation used to seal the chamber and prevent gases escaping through the breech. This may use a metal cartridge case that also holds the propelling charge, a configuration called "QF" or "quickfiring" by some nations. The alternative does not use a metal cartridge case, the propellant being merely bagged or in combustible cases with the breech itself providing all the sealing. This is called "BL" or "breech loading" by some nations.
A second characteristic is the form of propulsion. Modern equipment can either be towed or self-propelled (SP). A towed gun fires from the ground and any inherent protection is limited to a gun shield. Towing by horse teams lasted throughout World War II in some armies, but others were fully mechanized with wheeled or tracked gun towing vehicles by the outbreak of that war. The size of a towing vehicle depends on the weight of the equipment and the amount of ammunition it has to carry.
A variation of towed is portee, where the vehicle carries the gun which is dismounted for firing. Mortars are often carried this way. A mortar is sometimes carried in an armored vehicle and can either fire from it or be dismounted to fire from the ground. Since the early 1960s it has been possible to carry lighter towed guns and most mortars by helicopter. Even before that, they were parachuted or landed by glider from the time of the first airborne trials in the USSR in the 1930s.
In SP equipment, the gun is an integral part of the vehicle that carries it. SPs first appeared during World War I, but did not really develop until World War II. They are mostly tracked vehicles, but wheeled SPs started to appear in the 1970s. Some SPs have no armor and carry few or no other weapons and ammunition. Armored SPs usually carry a useful ammunition load. Early armored SPs were mostly a "casemate" configuration, in essence an open top armored box offering only limited traverse. However, most modern armored SPs have a full enclosed armored turret, usually giving full traverse for the gun. Many SPs cannot fire without deploying stabilizers or spades, sometimes hydraulic. A few SPs are designed so that the recoil forces of the gun are transferred directly onto the ground through a baseplate. A few towed guns have been given limited self-propulsion by means of an auxiliary engine.
Two other forms of tactical propulsion were used in the first half of the 20th century: Railways or transporting the equipment by road, as two or three separate loads, with disassembly and re-assembly at the beginning and end of the journey. Railway artillery took two forms, railway mountings for heavy and super-heavy guns and howitzers and armored trains as "fighting vehicles" armed with light artillery in a direct fire role. Disassembled transport was also used with heavy and super heavy weapons and lasted into the 1950s.

Caliber categories: A third form of artillery typing is to classify it as "light", "medium", "heavy" and various other terms. It appears to have been introduced in World War I, which spawned a very wide array of artillery in all sorts of sizes so a simple categorical system was needed. Some armies defined these categories by bands of calibers. Different bands were used for different types of weapons—field guns, mortars, anti-aircraft guns and coastal guns.

Modern operations: List of countries in order of amount of artillery (only conventional barrel ordnance is given, in use with land forces):

Artillery is used in a variety of roles depending on its type and caliber. The general role of artillery is to provide fire support—"the application of fire, coordinated with the manoeuvre of forces to destroy, neutralize or suppress the enemy".  This NATO definition makes artillery a supporting arm although not all NATO armies agree with this logic. The italicised terms are NATO's.
Unlike rockets, guns (or howitzers as some armies still call them) and mortars are suitable for delivering close supporting fire. However, they are all suitable for providing deep supporting fire although the limited range of many mortars tends to exclude them from the role. Their control arrangements and limited range also mean that mortars are most suited to direct supporting fire. Guns are used either for this or general supporting fire while rockets are mostly used for the latter. However, lighter rockets may be used for direct fire support. These rules of thumb apply to NATO armies.
Modern mortars, because of their lighter weight and simpler, more transportable design, are usually an integral part of infantry and, in some armies, armour units. This means they generally do not have to concentrate their fire so their shorter range is not a disadvantage. Some armies also consider infantry operated mortars to be more responsive than artillery, but this is a function of the control arrangements and not the case in all armies. However, mortars have always been used by artillery units and remain with them in many armies, including a few in NATO.
In NATO armies artillery is usually assigned a tactical mission that establishes its relationship and responsibilities to the formation or units it is assigned to. It seems that not all NATO nations use the terms and outside NATO others are probably used. The standard terms are: direct support, general support, general support reinforcing and reinforcing. These tactical missions are in the context of the command authority: operational command, operational control, tactical command or tactical control.
In NATO direct support generally means that the directly supporting artillery unit provides observers and liaison to the manoeuvre troops being supported, typically an artillery battalion or equivalent is assigned to a brigade and its batteries to the brigade's battalions. However, some armies achieve this by placing the assigned artillery units under command of the directly supported formation. Nevertheless, the batteries' fire can be concentrated onto a single target, as can the fire of units in range and with the other tactical missions.

Application of fire: There are several dimensions to this subject. The first is the notion that fire may be against an opportunity target or may be arranged. If it is the latter it may be either on-call or scheduled. Arranged targets may be part of a fire plan. Fire may be either observed or unobserved, if the former it may be adjusted, if the latter then it has to be predicted. Observation of adjusted fire may be directly by a forward observer or indirectly via some other target acquisition system.
NATO also recognises several different types of fire support for tactical purposes:

Counterbattery fire: delivered for the purpose of destroying or neutralizing the enemy's fire support system.
Counterpreparation fire: intensive prearranged fire delivered when the imminence of the enemy attack is discovered.
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.