Cliff notes version:
F/TR: .223 Rem or .308 Win, fired from a bipod and/or sling with a rear bag. Weight limit 18.15lbs as you pick the gun up off the ground. No limits on barrel length, twist rate, stock design, trigger weight, or action type. 'Unmodified' .223 or .308 is normally interpreted as 'not improved'; tight/match chambers with custom throating are common place in both NRA High Power, Fullbore, and F/TR (which has its roots in Fullbore). This is not a 'factory' class nor is it a 'Tactical Rifle' class, though many rifles of either type fit neatly within its rules. 'TR' stands for 'Target Rifle'...
F/Open: Caliber limit of .35, or as set by hosting organization or facility. Weight limit of 22lbs as you pick it up off the rests (does not include front rest or rear bag). Fore-end width limit of roughly 3" (76mm, which is just a fuzz under 3"). Basically no other limits on stock design, trigger weight, etc.
Solid contact with the ground is not allowed (like an Accushot monopod), nor a mechanically adjustable rear rest, nor one-piece rests. You can stack pretty much whatever you want under either end of the gun to achieve a level firing point, but there are defined limits on the size (width). Rifle must be fired 'from the shoulder', i.e. not 'free recoil'.