Anti-Tank Guns during World War Two varied radically by type, nationality, and intended target. Each combatant nation fielded a dozen or more designs throughout the war, each with their own pros and cons. So, like flotiste says, its hard to provide a more detailed answer to your first question with some kind of limiting factor, like type, calibur, or period.

Generally speaking though, as the war progressed guns got larger, could fire across longer ranges, and penetrate thicker armor. This was a result of several developments. First, anti-tank guns developed from infantry field guns, which were direct fire artillery of small size and low weight. These guns were great for moving into and out of battle by hand (by sheer grunt-power), and could support infantry attacks. They were also designed to fire AT shells of a small caliber, which was effective against lighter tanks. A good example would be the German Pak36 37mm gun, whose crew could push it into and out of battle by hand, and could fire a variety of shells, including AT shells.

The problem with these early guns was in the size of the projectile and the length of the barrel. Longer barrels and larger shells= more range and more destructive power. The Pak36 was small, and had a short barrel. That meant it had a limited range, and limited penetration potential. In battle, this meant that the Pak36 was too small to engage most targets beyond 50-100 yards, and was dubbed the "Door knocker" because its only effect against enemy tanks was to let them know an AT gun was around.

As the war progressed, the Pak36 was replaced by larger AT guns like Pak40 and the Pak43. The most famous German AT gun of the war wasnt even an AT gun! The German Flak 36 was an 88mm gun which was designed to hit bombers and aircraft at low and medium altitudes. However, it was also successful in an AT role due to its solid design and high velocity round. It also represents how guns changed as the war developed. Most AT guns moved towards longer barrels, which increased range and accuracy, as well as larger shells, which increased penetration and damage. Guns in the 75-90mm range, which featured longer barrels, were more capable than the smaller guns of the early war period. The German 88mm guns were capable of defeating almost any tank in the Allied arsenal at ranges of up to 1000m, and could kill a Sherman tank at 1500-2000m, an improvement over the limited capabilities of the Pak36.

The drawback of the largers guns was that they were harder to get into position. As you can tell by looking at all these guns, the crew who served them were often put out in exposed positions. They relied on speed, concealment, and infantry support to get their guns into position to fire on enemy positions. However, the heavier guns meant that it was harder for the crew to move the gun by themselves, and some of the larger models could only be moved by truck or tractor. This limited their flexibility, and made AT guns more immobile. For this reason, most armies also included a series of lighter guns, which could still be manhandled, but could do more than just annoy enemy tanks.

As for how they operated in combat, AT guns fired at specific targets. A commander would select the target, take range, and help traverse the gun. He would also specify the round (Armor Piercing, AP Composite Rigid, High Explosive, or High Explosive Anti Tank). The Loader would load the shell while the gunner put the gun on target. The commander would order the gun to fire, observe the results, and select a new target accordingly. By the end of the war, AT guns were purpose built to fire on tanks, but many guns also had the capability of firing on infantry positions and fixed fortifications, and were thus also expected to support infantry attacks. This would also provide the infantry with some protection if an enemy tank decided to show up.

Sorces:

John S. Weeks, Men Against Tanks: History of Anti-Tank Warfare

Stephen Napier, The Armoured Campaign in Normandy

Different platforms under the AT gun umbrella had different characteristics, but generally speaking they were accurate, and fired at specific targets.

One of the most ubiquitous AT guns was the Pak-38, used by Germany early in the war. It is sighted for ranges between ~300 and ~2500 meters, which gives you an idea as to the ranges it was expected to be employed at. As developments in armor necessitated increasingly larger calibers, the effective range increased accordingly. This table shows the armor penetration abilities of different German AT guns at various ranges out to 2000m.

Though AT guns could fire high-explosive which required less accuracy, they were primarily used in the direct-fire role, firing along relatively flat trajectories as opposed to plunging fire weapons like mortars and artillery that ware used (usually against infantry) to saturate target areas.

Anti-tank guns were also considered by tank crews to be more dangerous than enemy tanks, because they were much easier to conceal and were equivalent in terms of lethality. German tank ace Michael Wittmann stated that he was more proud of his kills made on AT guns than he was against enemy tanks.

(Taylor, Daniel (1999). Villers-Bocage Through the Lens. After the Battle. ISBN 1-870067-07-X.)

Another point specific to the German army is that late in the war, the quality of AT ammunition deteriorated as good steel was hard to come by, which affected both accuracy and armor penetration.

Each weapon has different accuracy. Is there one in particular you wanted to know about?