I heard two terms 'back country camping' and 'front country camping' but I don't know what the difference between the two is. How are they different.
Front country is when you go camping in an established campground that is usually reached by car. While backcountry camping is when you go camping in the backcountry getting to the non established campground by foot or horseback carrying your stuff with you.
So this would be an example of a front country campground with a fire ring and picnic table and a car.
and this would be an example of a backcountry campsite which I backpacked many miles into and there were no human created fixtures of any sort besides the trail.
Of course, you can have camps that fall between such as designated backcountry tent campsites in National Parks with provided bear storage.
Charlie has given you an excellent summary of the difference between the two, focused on the difference in exertion and time to reach them, and the amenities to be found at each.
There is another important difference: front-country camping gives you very little, if anything, of the feeling of wilderness. You are in a campground with many other people, usually within sight of their tents and cars and food, and within sound of their conversations and whatever noise they are making. You have very little privacy, often no privacy. If you have forgotten something essential, you just hop in your car and go to the nearest store, which is usually pretty close.
In backcountry camping, it is not unusual for there to be no one at all anywhere near your campsite especially if you are on a little-used trail or have left the trail and are hiking cross-country. If you hike cross-country, you may see nobody at all for days once you have left the trail. There is nobody to help you if you need help, and nowhere to get whatever you forgot. You are in wilderness, and know it from the vast silence, broken only by the singing of coyotes, the magnificent vistas empty of human presence (and litter), and the night sky with more stars than you have ever seen before, unobscured by artificial lights.
You need experience to do backcountry camping safely, which you acquire gradually. First do some front-country camping to make sure you thoroughly understand your equipment. Then, go one day in along a trail, then, several days along a trail, then some modest cross-country backpacking. You probably should go with a group on your early trips.