- hardtail - front suspension only
- rigid bike - no suspension fork or other shock absorbing features of the frame
- full suspension - front and rear suspension
If you're just starting out in mountain biking, you'll have plenty of good times riding and do so at a reasonable price by getting a front suspension mountain bike. A mountain bike with front shocks will handle almost all the riding situations you will encounter, until you start really getting off singletrack riding, take on riding off boulders, and pushing your abilities to new levels. Or, should you branch out to a specific type of mountain bike riding, such as the ever-fun downhill style of riding, in which case you'd want to buy a more specialized bike for downhill riding. For a beginner rider, you can tackle off-road riding right away with a front suspension mountain bike, which by the way, also goes by the name hardtail.
The next step up in mountain bike buying is to get a full suspension mountain bike. In this case, you'll have the front fork shocks as before, but also a rear shock system that is integrated with the frame in some way to provide shock absorption for your derrier. These mountain bikes are called dualies and are intended for advanced or hard-core mountain bike riders who are navigating much more difficult obstacles and terrain. Generally, a quality full suspension mountain bike is going to cost more than an entry level hardtail, though there certainly exist dualies which are priced well for budget-minded bikers. Bear in mind, as always, that you get what you pay for. Take a close look at the rear suspension solution on whatever bike you choose, evaluating its ability to dissipate shock and how well constructed it is.
As is true with any type of bicycle, the frame is the centerpiece of a good mountain bike. When it comes to ride quality and the other intrinsic characteristics of your bike, it all comes back to the frame. Bike frames can be hand-made, which means either welded by hand or having carbon sheets layed up manually to create the frame form, or machine made, which involves computers and robots programmed to make the necessarily welds on frames as they are precisely placed in frame holders. True bicycle frame craftmanship comes from a real person hand making a frame, and that is the ideal. Of course, it also means a higher ticket price. With modern bike frame technology, CAD design, and excellent quality control, you'll do just fine with a machine-made bicycle frame, which in fact is what you're going to find in most bicycle stores.
Most mountain bike frames are built using what is called a double triangle geometry. This is just the technical term for the bike frame you're familiar with and see every day, a diamond style frame. However, when it comes to dual suspension mountain bikes and downhill bikes, the double triangle is modified to accommodate the rear shock design, whether that be in the seat tube, at the juncture of the seat tube and down tube, or some other location.

Rocky Mountain Trailhead - hardtail mountain bike - diamond frame
So that's a Klein Mantra pictured below. Contrast that monoque frame design with the mountain bike shown below, which has a rear suspension. You can see the different frame geometry required in order to fit the rear shock as well as to allow for the movement of the back section of the bicycle. This is just one dual suspension mountain bike design, and you will certainly see many other creative variations. Mountain bike manufactuers, fueled by rider input, are always innovating on frame designs so as to meet the demands of those mountain bike riders.

Klein Mantra Pro - unique frame design
Another frame characteristic often seen is a sloping top tube. This design element is desirable in some cases, depending on the rider height, as it provides more clearance between the bike's top tube and the rider's crotch when standing over the bike. It's easy to understand why this design is nice to have - as a mountain bike rider, you're going to crash at some point, and having that clearance can mean the difference between laughing off a wipeout and crying in groin-induced pain!
A mountain bike without a shock absorbing front fork is referred to as rigid. There are some cases in which having a rigid mountain bike makes sense - but not many. There are some hard-core, purist mountain bikers who can descent through crazy hard rocky terrain that beats up other riders, yet do this on a regular fork. Why? Because it's that much more bad to the bone to navigate tough trails, rocks, or boulders without the aid of front suspension. The other case is if you are going to use your mountain bike primarily for commuting or on smooth, packed surface singletrack or trails. In this event, the excess energy lost in the shock actuating is just not worth having it. A mountain bike with front suspension, all in all, is your best bet, since it will enable you to ride under regular street conditions as easily as off-road. And in the event that you should roll through a small pothole, or strike a tree branch on the ground, the front fork shock will eat up some of that unexpected jolt - and may just keep you upright on your bike.
The only other consideration here would be money. If you are on a really tight budget, then going with a mountain bike with a rigid fork will indeed knock down the bike price by at least $100, and likely more than that. There are many mountain bikes out there in the sub $300 price range - good bike brands that will last, not department store bikes - that will leave you funds to get other items you'll need, like a bike lock and bike lights and other riding gear. For example, consider the Specialized Hard Rock, a mountain bike staple that has been around for many iterations, and has a rigid fork. It is better for off-road riding than most other bikes in its price range, even those that do sport front suspension. That is because most suspension forks on lesser bikes are not that well built - sure, they add some comfort for off-road riding, but they don't have the rigidity and damping capability of better forks, which are helpful for letting you keep control of your mountain bike while riding in rough conditions. So the Hard Rock Rigid is an option to get you started in mountain biking, and then you can upgrade your bike later by purchasing a suitable front suspension fork some day.

Marin Kentfield - rigid mountain bike - with suspension seatpost
Good quality rigid mountain bikes are now harder to find, what with the popularity of front suspension MTBs. There are nice bikes to be found, with a little effort - for example, some manufacturers call rigid bikes "urban bikes" or, in some cases, "commuter bikes". Though, a commuter bike really does have features which should, by definition, differentiate it from a mountain bike, such as rear rack bolts and fender connection points.
You can find mountain bikes equipped with suspension seatposts. This is usually accomplished using elastomers and embedding them just below the seatpost head, thereby allowing for some vertically give in rough riding conditions. You can always buy a front suspension mountain bike and then upgrade your seatpost later. A suspension seatpost won't have nearly the shock absorption capability as compared to the rear suspension of a dualie, but it might just provide enough shock-eating comfort for your style of riding and for your regular rides.
Disk brakes were once a higher end bike option, but now the technology has trickled down to where you can expect to find disk brakes on almost all the mountain bikes you look at. They won't muddy up as easily as cantilever brakes, thereby assuring you of braking power when you need it. Since the price level on disk brakes has dropped so dramatically, you should opt for the superior braking power of disk brakes on your bike. Bear in mind, they really do brake with force - if you can get by with cantilever brakes, and you prefer the operability and maintenance of them, then by all means do so. Disk brakes seem to modulate a bit better than other brake types, which translates to finer feathering of braking power when you're in tight situations on the trail or navigating rocky terrain.