Full Suspension
Full suspension (a.k.a. "dual suspension" or "soft tail") bikes include a front
suspension fork and rear suspension. Subsets of full suspension designs include Single Pivot, Four Bar and VPP.
Single PivotSingle Pivot suspensions are normally found on cheaper full suspension bikes. They will have more bob than a VPP or 4 Bar suspension.

Four BarThe Four Bar (and its variant the Faux bar) use several linkage points to activate the shock. A 'true' Four Bar will have a pivot behind the
bottom bracket, one in front of the rear
wheel dropout (this pivot being the venerated "Horst Link"), and one at the top of the
seat stays. A Faux Bar is similar, but will have a pivot above the
dropout instead of in front of the
dropout (i.e. no Horst Link and no patent problem). The importance of this one pivot is heavily debated with supporters on both sides of the debate. Four Bar designs include Norco "VPS" bikes, almost all Specialized bikes, Kona, Ellsworth, KHS, Turner (although their new '06 Flux has lost the Horst link), and Ventana.

VPPVPP stands for "virtual pivot point." VPP technology is important because it allows the rear wheel's movement to be vertical, rather than circular (similar to a single pivot point suspension). Why don't you want your rear
tire to move in a circular motion? The simple answer is that it decreases bob.
The longer answer is that if the rear tire's pivot motion can be described by an arc, and the
tire exerts torque on the pivot system, then the whole system will begin to flex when force is exerted and the rear shock absorber will start to compress. If a
tire simply moves up and down, then the tangential force applied by the torque of the
tire won't be capable of moving the whole system.
Technically, VPP technology was invented and patented by Mongoose, and then later when Mongoose was acquired by Pacific Cycles (the primary supplier of Walmart, other brands include "Next," "Roadmaster," and "Schwin"), the patent wasn't used in favor of producing cheaper bikes. Now, companies are developing their own proprietary technologies to get around Mongoose's patent.

Suspensions
Rear SuspensionPerhaps because front suspension has been easier to implement and more readily adopted, it is often assumed that rear suspension is sometimes synonymous with full suspension.

Suspension SeatpostMany hardtail bikes are retrofitted with suspension seatposts to take the edge off some of the bumps encountered on the trail. Unlike rear suspensions, seatposts don't suffer from any problems with rear
tire bob. Pictured below is a seatpost with an elastomer dampener, and while this is a popular choice some seatposts use more conventional piston shocks.

Terminology
BobBob refers to how a suspension (usually rear) responds to a rider's pedaling. It is the up and down motion caused by the rear
tires when you start pedaling hard. Bob is usually an undesirable characteristic as it robs power from pedaling. This tends to limit the amount of power the rider can create, which increases the difficulty of climbs. Many suspension systems incorporate anti-bob or "platform" dampening to help eliminate bob.
Compression DampingCompression damping refers to systems that slow the rate of compression in a front fork shock or rear shock. Compression damping is usually accomplished by forcing a hydraulic fluid (such as oil) through a
valve when the shock becomes loaded. The amount of damping is determined by the resistance through the
valve, with a higher amount of damping resulting from greater resistance in the
valve. Many shocks have compression damping adjustments that vary the resistance in the
valve. Often,
lockouts function by allowing no compression.
HardtailHardtail bikes do not have a rear suspension, and they may or may not have a front fork (a fully rigid bike is a hardtail). Hardtails are lighter and cheaper for a good bike (because you don't have to buy quite as complicated of a
frame in addition to another shock), and pedaling is more efficient because they don't bob. However, in bumpy terrain at fast speeds, their rear
tire can come in contact with the ground less than a soft tail's. They are also less comfortable.
LockoutLockout refers to a mechanism that disables a suspension mechanism to render it substantially rigid. This may be desirable during climbing or sprinting to prevent the suspension from absorbing power applied by the rider. Some
lockout mechanisms also feature a "blow off" system that deactivates the
lockout when an appropriate force is applied to help prevent damage to the shock as well as rider injury under high unexpected loads.
PreloadPreload refers to the tension applied to spring components before external loads, such as rider weight, are applied. More preload makes the suspension stiffer and less preload makes it softer.
SagSag refers to how much a suspension moves under just the
static load of the rider.
Soft TailSoft tail bikes include any bikes that have a built in pivot point, which moves the rear
wheel. TravelTravel is the maximum amount of movement that a suspension system is capable of. Three to five inches of travel is average on most full suspension bikes.
Downhill racers and freeriders use up to nine inches of travel.