Both French and German infantrymen were armed with a breech-loading, bolt action rifle utilizing a paper black powder cartridge. The German and French variants (Dreyse "needle gun" and Chassepot model 1866, respectively), were a marked improvement on the muzzle loading rifle-muskets they replaced, and the Franco-Prussian war was the first major conflict in which they were fielded by both sides. The Chassepot was generally considered the superior weapon, but the advantage conferred to the French strategically was negligible. I admit I don't know a great deal about the sidearms of the conflict, although I would conjecture that cavalry troopers were most likely issued some sort of standard revolver. The French had retired the Lefaucheux m1858, a pinfire revolver, five years before the war broke out, and it was never officially issued to the army, although marines, as part of the navy (which did make use of the weapon) may have carried them. In contemporary photographs and illustrations, infantry officers are almost never carrying pistols, suggesting that they weren't issued sidearms at all, while most cavalry and artillery officers have one. Perhaps officers brought a wide variety of pistols as part of their personal effects. I just don't know enough (or have enough time) to give you a definitive answer.

Artillery demonstrated a greater disparity in arms and organization. In 1870 the French were transitioning from the La Hitte artillery system (consisting mostly of rifled muzzle-loading cannon, and designed to retrofit old pieces to fire explosive shells), which had been in place since 1858, to a newer system built around the guns designed by Reffye. Officially, the standard field gun of the French army was the "Canon de campagne de 7 de Reffye modéle 1870," an 85mm breechloading rifled cannon. At the outbreak of hostilities, however, many French units hadn't been issued the new pieces, and had to make do with the old La Hitte models. In practice this meant that the balance of French formations were supported by mostly muzzle-loading, rifled brass 12 pounders. These would have mostly been retrofitted "canon de 12" pieces. The French also deployed the Reyffe "Mitrailleuse," a forerunner of the machine gun, with their artillery batteries. Their intended purpose was to supplement anti-infantry firepower beyond rifle range and, as Reyffe himself said, "compensate the insufficiency of grapeshot" at ranges beyond 1000 yards. The weapon was deployed in accordance with French operational doctrine, which stressed the use of massed rifle fire from prepared positions, and encouraged French officers to rely on their superior infantry firepower by generally assuming the defensive, and using entrenchments or cavalry maneuvers to resist or neutralize enemy artillery fire.

Prussian doctrine, on the other hand, was designed for the offensive, and their artillery arm reflected as much. The German armies' main field gun, the Krupp C64, was a steel, breech-loading 8cm cannon. It had an effective range of about 3.4km, which greatly outdistanced the majority of French tubes (the old retrofitted 12-pounders, for example, could only manage about 1800 yards). German siege artillery, including the famous 1000-pounder Krupp siege guns used to shell Paris into submission. Krupp, by the way, refers to the Krupp family ironworks, then located in Essen, that produced a huge portion of Prussia's military hardware. Krupp anti-balloon guns were also the first purpose-built antiaircraft weapons.

Mortars, as they're employed in modern warfare (i.e. man-portable, deployed in infantry formations) didn't really exist in 1870. Massive siege mortars, on the other hand, were prevalent in both armies.

Edit: Forgot my sources!

"The Franco-Prussian War: The German Conquest of France in 1870-1871." Wawro 2003

"Tactics in the Franco-Prussian War From Opening Shots to the Battle of Sedan" Johnson

I used wikipedia for dates concerning the French artillery systems (since I didn't want to comb through Wawro looking for them) and for the stuff about the Lefaucheux revolver.

Does anyone know what specific models of Pistols, Revolvers, Mortars, or Artillery Pieces were used during the conflict?

I don't know of revolver designs, but the French utilized rifled muzzle-loading artillery pieces, of the La Hitte system. The Germans in turn utilized breech-loading artillery pieces, notably the Krupp C/64 77mm (4 pounder) piece and the larger c/67 6 pounder piece. The French had some Reffye and Montigny Mitrailleuses, which were rapid firing volley guns rather than machine guns as they are often described. If you're looking for good books on the subject of the war, Geoffrey Wawro's history of the war, and Dennis Showalter's history of the Wars of Unification are solid.