A rigid frame bike has no suspension. It's actually difficult to find a rigid frame mountain bike, although a few manufacturers do offer one or two rigid models. Even low-end mountain bikes usually come with a front suspension, also known as a suspension fork. A bike with a suspension fork but no rear suspension is known as a hardtail. If a bike has both a front and rear suspension, that's known as a full-suspension bike.
Which is the right one for you? Each suspension element adds complexity and weight to the bike. A suspension fork trades this for increased control and comfort. A rough ride can take a toll on your wrists and arms if they're absorbing every impact. The control aspect is even more important. Without a front suspension, the bike's front tire will have a tendency to bounce off of obstacles like rocks or branches. When your front tire is bouncing, you can't steer. Plus, if you hit something on an angle, it can send your front tire bouncing to either side. That can lead to serious crashes. Unless you plan to ride exclusively on groomed trails or bike paths, a good suspension fork is crucial.
We've mentioned front suspensions, but do you need a rear suspension? Not necessarily. A rear suspension adds more weight, and requires a different frame, which makes it difficult or impossible to add a rear suspension to a hardtail bike. You could add a shock absorber to the seat post, which will add comfort but won't help you stay in control in rough terrain, which is really the point of a suspension. Many mountain bikers swear by rear suspensions -- in fact, they're the standard for competitive riders.
This leads to the question: Which type of rear suspension is the right one? Mountain bike rear suspensions have undergone extensive development in the last 20 years. They are difficult to design because the suspension is not just affected by impacts with obstacles, but also by the rider's weight, plus acceleration and deceleration forces. The unified rear triangle (URT) design common several years ago is now falling out of favor, as it tends to react poorly to acceleration, and becomes almost entirely ineffective if the rider stands on the pedals. Most bikes today are made with a raised low pivot (RLP) design, which uses an independent swingarm. There are endless variations on the basic RLP suspension, with each manufacturer developing its own particular type. Only experience and test rides will tell you the type that will work best for you.
Now you're hurtling down a mountain trail on your bike. How do you stop? Find out in the next section.