The aspect ratio is an airplane's wing's span divided by its standard mean chord (SMC). It can be calculated more easily, however as span squared divided by wing area.
Planes with a higher aspect ratio are generally more efficient because they suffer less induced drag and their wingtip vortices do not affect the wing as much. However, the long wings mean that the plane has a greater moment of inertia about its longitudinal axis and therefore cannot roll as quickly and is less manouverable. Thus, combat aircraft and aerobatic planes usually opt for shorter wingspans and lower aspect ratios to increase manouverability.
Long, narrow wings with very high aspect ratio, being very efficient, allow aircraft like the U-2 to travel at extreme altitutes, but are hit by the shockwave at supersonic speeds. It's been proposed this type of wing might be appropriate for Martian aircraft as atmospheric pressure ground level is roughly equivalent to 100,000' on Earth.
Delta wings are the other extreme. With a small span for their area they have a very low aspect ratio. Their shape makes them ideal for supersonic flight, but are very inefficient at high angles of attack, such as at takeoff, landing and in turns. Despite the need for enormous thrust at high angles of attack to maintain energy, delta aircraft can continue past the point of stall for conventional wings and are often favoured in fighters for rapid turning. Variations beyond the pure delta include the cranked (Avro Vulcan), compound (Shuttle) and ogee (Concorde) deltas. All improve dynamics in a specific region, tailored for the aircraft's use.
A longer definition at Wikipedia: