Some general terms:
RV Park - private park, almost always with full hookups, often minimal space between sites, frequently located within walking or short driving distance to restaurant(s), stores, etc. Usually offers laundry facilities, shower facilities, and often a convenience store. Might have a golf course nearby.
Private Campgrounds - usually a resort style campground with planned activities for the kids and adults, convenience stores, swimming pool, etc. Set in a more rural setting than an RV Park and may have cabins/yurts for rent and/or a section for tents. Often has at least electrical and water at the sites and a dump station nearby.
Public Campgrounds - these are the federal, state, and local government owned campgrounds and vary the most. They can be extremely difficult to get reservations for (National Park campgrounds) or they can offer free boondocking locations (Bureau of Land Management - BLM). They vary greatly on amenities such as hookups, convenience stores, restroom facilities, planned activities, etc. They vary greatly as to location and closeness to urban and/or tourist centers.
Of the public campgrounds, the most common choices are:
National Park Campgrounds - these are the most difficult to get into and, unless you can go in the off season and on a weekday, you will most likely find that reservations are needed. In each park, you will usually find at least one campground with hookups, but it will fill up the fastest (again, reservations being needed more than likely). There is usually at least one campground that is "first come first served" but you will have to be there early to get a spot.
National Forest/Grassland Campgrounds - these vary depending on the region. In the west, these are in the mountains away from urban/tourist areas and have no hookups. Bathroom facilities can vary from flush toilets and showers to pit toilets only. They are usually fairly inexpensive and are preferred by campers.
Corps of Engineers (COE) - these are often built around man-made reservoirs, etc. They cater to outdoor summer activities (boating, skiing, hiking, fishing, etc.). Vary on hookups and facilities. Usually fairly inexpensive.
Bureau of Land Management (BLM) - these lands are great for boondocking (see below).
State Parks - these vary GREATLY depending on the state. Some states are free, some charge as much as $70 a night; some have full hookups, some have nothing; some have all sorts of activities, some have nothing; and so on; some are minutes away from urban areas, some are hours away... Ask about a particular state's system for specific information.
City/County Parks - these are often located right in the middle of a town and can be a jewel in a haystack. Rules vary between cities/counties but are often more relaxed than their fellow state parks.
Camping Terms:
Full Hookups - electricity, water, sewer on site; often with cable/internet on site.
Dry Camping - no hookups on site.
Boondocking (aka Dispersed Camping) - dry camping outside of a developed campground. This is usually free camping. BLM lands mostly allow you to boondock for a certain number of days. There are some rules about how far off the road you can go, specific areas that are prohibited, etc., so always get with the local office for any pertinent information.
Overnighting - this is what folks do when they park at a Walmart or Cracker Barrel or rest stop for the night.
Stealth Camping - this is "usually" illegally parking on a street overnight and trying not to be noticed by the locals (hence the "stealth").
Hope that helps. Folks will interchange these terms a lot but if you keep them separated in your head, you will have an easier time figuring out what you want to do and how to ask for help in doing it.