The hardtail mountain bike features front suspension but no rear-wheel suspension. Other types of bikes include full-suspension mountain bikes and rigid bikes.
Hardtails offer a nice middle ground, providing front suspension to eat up bumps on rough downhills, but allowing for a rigid rear wheel that provides stability and traction on extended uphill climbs. A full-suspension bike can bounce up and down when climbing and limit forward momentum. Hardtails weigh less than comparably priced full-suspension bikes, given the lack of the additional rear suspension hardware. They're also easier to maintain and more durable.
Hardtails are best for lighter cross-country use that doesn't involve large drop-offs, huge bumps and crazy downhill speeds. Hardtails are not well-suited for freeride or downhill mountain biking. They're an excellent option for recreational riders who ride smoother terrain and don't seek to bomb at high speeds.
Obviously, hardtails won't provide the same cushioned ride of a full-suspension bike. On rough terrain and fast downhills, a hardtail can feel particularly choppy.
Most any mountain bike company offers a line of hardtails. Hardtails are cheaper than full-suspension models and are available in a wide price range.
The decision of hardtail vs. full suspension, and rigid for that matter, is largely user preference, assuming you're a cross-country rider. Consider how rough the terrain is and the size of obstacles that you'll be rolling over. Determine how important a smooth, bump-free ride is in comparison to faster, more efficient ascending. Discuss it with local mountain bikers and shop staff in your area, but take advice with a grain of salt because most riders have a preference one way or the other. If you do decide with a hardtail, you'll get a bike that is superior in nearly every other way over a similarly-priced full-suspension bike (weight, components, materials, etc.).