You can camp for free in all unimproved areas of US national forests that aren't designated wilderness areas or otherwise designated as off limits for dispersed camping. This is something like 98% of the forests. The exact rules can vary a bit from place to place, so always check with the ranger station in that area (don't rely on the website!) for anything special.
You usually need to have a permit if you're parking your car at a trailhead, though.
But in general, the rules are:
Leave no trace. This means: no litter, bury all your poop (but no toilet paper) in holes at least 6 inches deep, use spots that have already been used for camping before when possible, etc.
Don't damage anything.
Don't cut live trees or other plants.
You can collect and burn all the firewood you want, but you cannot take any out of the forest.
Generally speaking, do not use fireworks or firearms.
Fires must be completely extinguished if there is no person around. Completely extinguished means that you can stick your hand into the ashes without discomfort.
Store your food in a way that prevents animals from getting to it. Do not feed any animals, not even squirrels or birds.
No camping within 100 feet of any water sources, trails, or improved campgrounds.
You cannot stay in the same area (I think, but I'm not sure, that "the same area" means within 50 miles) for more than 14 days within a 31 day period.
Do not build any structures.
If you leave your campsite unattended for more than 72 hours, the forest service deems it abandoned and may disposed of everything in it.
No amplified sounds (music players, PA systems, etc).
Also, for most of the "don't do this" rules, there is an implied "without a permit". For instance, you can remove wood from the forest if you have a permit to do so.