Full-suspension mountain bikes have a shock and swingarm mounted on the back of the bicycle frame and a suspension fork up front. A hardtail does not have the rear shock and swingarm, just the suspension fork up front.
Which configuration to choose has been a source of heated debate since the early days of the sport. More and more, it seems fully suspended bikes are gaining acceptance as the preferred trail rig. That being said, hardtails and full-suspension bikes each have their pros and cons.
Weight. Generally speaking, hardtails are lighter than full suspension bikes at the same price point. As evidenced by the mix of hardtail and full-suspension bikes on the pro XC race circuit, it is debatable weather this weight savings equates to quicker lap times. Some argue that the bump absorption allows the rider to sit and pedal through rough terrain that would force a hardtail rider out of the saddle.
Rigidity. The pivots and bushings/bearings of the full-suspension bike can mean more flex. In most cases, the hardtail will provide a more efficient peddling platform on smooth terrain. The actual rear suspension itself can be a power leak if there is pedal induced bobbing. In recent years, however, many of the manufacturers have found ways to make the rear suspension know the difference between a bump and a strong pedal stroke from the rider, making this less of an issue than it was with the bouncy early full-suspension designs.
Value. you will almost always be able to get more bang for your buck with a hardtail. If you examine a hardtail and full-suspension bike at the same price point, you will almost always find that the hardtail is spec'ed with nicer components.
So, which one is right for you? It really depends on what type of riding you are looking to do. If flat dirt, or paved trails are your bag, than you may get more value out of a hardtail. If performance trail riding on advanced, technical trails is more your speed, than you may want to consider the pros and cons a bit more closely before pulling the trigger.