As the name suggests, dual-suspension or full suspension mountain bikes have suspension in the front and rear of the bike. The suspension systems allow for 'travel' which is mountain bike terminology for the amount of suspension movement available. The suspension helps absorb the impact from the trails, reducing the impact on the rider and improving traction and comfort. The suspension helps keep the wheels on the ground for traction, as a result, the rougher the trail, the more suspension required. Typically travel will range from 80 mm - 200 mm front and rear, depending on the intended purpose of the bike.
Downhill bikes for example are built to go down steep, technical tracks as fast as possible, so have large amounts of travel to provide additional traction and support. At the opposite end, cross country bikes need to be light and efficient to pedal and so will typically feature around 100mm of travel. Many suspension systems allow riders to 'lock out' the suspension which effectively renders the shocks inactive, consequently reducing movement and making the bike stiffer which is better for climbing back up to the top of the mountain or riding on the road to the trail head.
A quick guide on suspension
The front suspension is located in the forks and is relatively simple in comparison to the rear suspension which comes in a variety of forms.
Suspension in the fork is provided by a spring which 'bounces' open after it has been compressed from hitting an obstacle along the trail. The spring can be either air, coil or both. Air springs are light weight and highly tuneable using a shock pump. Coil springs are used on lower cost bikes and also feature on downhill or gravity bikes. Made from either steel or titanium, the coil springs are extremely durable, providing a more responsive feel than air springs due to a lack of tight seals.
Rear suspension comes in many forms. All try to achieve similar things, which is to isolate braking and pedaling forces from the suspension action. The Four Bar system is perhaps the most common and features a chainstay pivot, a pivot behind the bottom bracket and a pivot on top of the seat stay which is connected to a leveraged shock linkage. The Faux Bar system is very similar to the Four Bar but with pivots on the seat stays. A variation on the Four Bar is the Virtual Pivot Point (or VPP) and the DW-Link which both use twin links to isolate the rear end. A Split Pivot has a rear dropout pivot placed in line with the rear wheel axle, and is virtually the same as Trek's Active Braking Pivot (ABP). A single pivot suspension system is the most simple option featuring a swing arm and one main pivot located above and forward of the bottom bracket. The Floating Drivetrain so called because the drivetrain is positioned on a link between the front and rear triangles is another popular system.
All of these systems have pro's and con's, and is sometimes dependent upon the manufacturer as to which system you'll get.