What permit do I need?
Please note that I have only backpacked in California, so I am only experienced with California requirements. I also believe the best thing you can do is call a ranger in the area you are going ahead of time to make sure that there are no last minute surprises you could have planned ahead for. The rangers are an excellent resource for everything from weather and fire conditions/restictions, to what your best bet is for catching fish. They enjoy helping out, so use them.
Some type of permit is required for most if not all National Forest, Parks and Wilderness areas. These are the areas that most backpackers frequent. For the most part you will need only a fire permit to camp in the National Forest Areas. These are free and can be obtained from any ranger station. They are good until December 31st of the year you get it. So what if you do not plan on building a fire (good for you!), well you still need one if you plan on using a stove. So yes even if you are not making a fire you will need a fire permit.
In wilderness areas you will need a wilderness permit to stay overnight and sometimes even one for day use. These are good for camping in very specific areas (sometimes down to an established campsite number). Most of the time you will not need to get a fire permit and a wilderness permit. (Usually the wilderness permit will tell you if you can or cannot have a fire.) Check to see what the requirements are in the wilderness you are headed to. Some popular trailheads (e.g. Desolation Wilderness, Yosemite) will have quotas that once met effectively close a particular region for camping. The way to work this is to get your permit in advance, go off-season, or go during the week. (I try to avoid crowded areas when possible, so I do a lot of Sunday/Monday trips lately.) Sometimes theyonly allow # many people per lake or per geographic region (it might include a few lakes in an area). I have also seen where they allow so many people per trailhead. This is another reason to call ahead so you do not find out the lake you want to camp near is already "full". I personally like the fact that they are limiting the amount of backpackers around popular spots. This is one way to preserve that region and make sure that we all feel like we are in the wilderness (not camping on top of each other).
Most wilderness permits cost so much per person per night. It usually costs a little extra to buy them ahead of time (over the phone with a credit card). If you do buy your permit ahead of time they will mail it to you (unless you bought it less than a week before your trip). Sometimes buying them ahead of time makes more sense if you are not traveling that close to the ranger station. Having the permit already gives you the option of heading directly to the trailhead.
National and local parks can have varying policies on overnight stays and even day use requirements, so I would recommend calling or going online for specific requirements.