When most people hear the word camping they usually picture sleeping in a tent out in the woods at a spot you can drive up to. This is a great way to camp, a few of the benefits:
Privacy: Usually when you camp on your own you'll be the only one around, or there may be people a few hundred yards away. That gives you a lot more privacy than a campground where your neighbors may be just 10-20 yards away. You'll have more peace and quite from others, and you won't have to worry about providing peace and quite to others.
Freedom: This goes along with the added privacy, but you'll be able to walk around and explore the area around your campsite.
Free: Camping on your own is usually free since you are not having to pay to have someone else maintain a campground. National Parks and some national forests do require back country permits.
The perks provided by camping on your own do come with a few drawbacks that you need to be aware of.
Harder to find: A campground will have directions on how to get there, but if you are camping on your own you need to find a suitable site on your own.
No reservations: Even if you already know where you are going and have a great site in mind, it's possible someone else had the same idea and got there first. Since you can't make a reservation on a free, public use campsite you need to be prepared to keep searching.
No supplies: If you are camping out in the forest there won't be any running water, restrooms, or fire pits. You'll need to be prepared with everything you need before you go. If you are camping near a stream or water source, make sure you have a good water purifier, you should never drink water straight from a stream.
Where can you camp? You can usually camp anywhere in a national forest, but make sure you are aware of any fire restrictions before you go. Also, it's important to clean up your camp after you leave and not leave any trace, being especially careful to make sure any fires are thoroughly extinguished. Some national forests do require a back country permit.
National Parks are a bit different, and you can usually camp in isolated back country sites if you get a permit at the park visitor center.