There are two forms of artillery in-game: Standard (cannon) artillery and rocket artillery.
Standard artillery takes the form of variously-sized howitzers such as the German leFH 18 or American 105mm M2. They operate very similarly to anti-tank guns; when requisitioned they are in towed form and must be moved into a location, deployed, and constantly supplied from nearby ammunition stores.
Standard artillery are best used at as long a range as possible, to prevent them easily being spotted and attacked by enemy forces. While they are capable of directly firing against nearby targets, their slow rotating speed makes it difficult for them to track anything moving. At long range, artillery can provide a steady stream of damage against a target area, killing enemy troops, destroying enemy structures and wreaking havoc with his logistics support.
Individually, artillery is rather ineffective owing to its slow reload time and lack of pinpoint accuracy; having a single artillery piece try to hit a specific location (such as an enemy machine-gun nest) will take quite some time. Therefore it is always better to group artillery into batteries so that their combined fire will saturate a target area.
Self-Propelled Artillery has the obvious advantage of vastly increased mobility, as they do not require any sort of deployment in order to operate -- they can drive to an area and immediately begin engaging the enemy, and can instantly withdraw to another location if they come under attack. This can, of course, make them harder to keep constantly supplied, so it is best to keep mobile halftracks near them whenever possible.
The key differnce is the cost, however, as mobile versions of artillery usually cost about 3 to 4 times as much as deployable ones. Yet the advantages are that you can focus a sudden barrage of fire in almost any point on the map, as well as counter enemy batteries without getting into any damger yourself. Mobile artillery cannot get pinned down, even by other artillery, but can be targetted by AT guns and tank destroyers.
Above: A battery of US artillery causes relatively light but continuous damage to an area.
The Germans and Soviets both field rocket artillery. The German variant is towed, and therefore has the same weaknesses as towed standard artillery. The Soviet version, however, is based on a mobile (but highly vulnerable) truck.
Rocket artillery has several advantages over standard artillery. Its main advantage is the ability to quickly unleash a barrage of missiles against its target in a fraction of the time it would take an entire battery of standard artillery to do the same. While less accurate, the sheer amount of firepower unleashed on the targeted area is sure to cause extensive damage.
Along with this advantage, rocket artillery has a significant disadvantage, namely in the high amount of logistics they use to keep themselves firing. Just as they pour out an immense amount of firepower in a short period of time, so too do they use up an immense amount of logistics supply in a similar short period. Despite this, the power of a battery of rocket artillery can often win games by completely devastating an enemy's base or defensive line.
Above: Several rocket artillery fire salvos that produce intense damage over an area in a short amount of time.
The Japaniese can field a small rocket mortar as well. It's difference is that it only fires a single, somewhat accurate missile that does impressive damage. The key difference with this mortar is that it is available at early stages of the game and can fire smkoe missiles as well as regular ones.