Wave drag is caused by the formation of shock waves around the aircraft. Shock waves raditate away a considerable amount of energy, energy that is "seen" by the aircraft as drag. Although shock waves are typically associated with supersonic flow, they can actually form at much lower speeds at areas on the aircraft where the Bernoulli effect accelerates local airflow to supersonic speeds over curved areas. The effect is typically seen at speeds of about Mach 0.8, but it is possible to notice the problem at any speed over that of the critical mach of that aircraft's wing. The magnitude of the rise in drag is impressive, typically peaking at about four times the normal subsonic drag.
If the problem of wave drag is caused by the acceleration of air over curves on the aircraft, the solution is, obviously, to reduce the curves. However this is not always easy, for instance, a wing generates lift at subsonic speeds primarily due to the curvature on the leading edge of the wing. Things are somewhat better for fuselage shaping, but simple things like a cockpit canopy or smoothing off the metal around an air intake can create additional "hot spots".
When the problem was being studied, wave drag came to be split into two – wave drag caused by the wing as a part of generating lift, and that caused by other portions of the plane. In 1947 studies into both problems led to the development of "perfect" shapes to reduce wave drag as much as theoretically possible. For a fuselage the resulting shape was the Sears-Haack body, which suggested a perfect cross-sectional shape for any given internal volume. The von Kármán ogive was a similar shape for bodies with a blunt end, like a missile. Both were based on long skinny shapes with pointed ends, the main difference being that the ogive was pointy on only one end.
These research projects were quickly put to use by aircraft designers. One common solution to the problem of wave drag due to the wings was to use a swept-wing, which had actually been developed before WWII and used on some German wartime designs (none of which saw service). Sweeping the wing to the rear makes it appear thinner and longer in the direction of the airflow, making a "normal" wing shape closer to that of the von Kármán ogive, while still remaining useful at lower speeds.
Fuselage shaping was similarily changed with the introduction of the Whitcomb area rule. Whitcomb had been working on testing various airframe shapes for transonic drag when, after watching a presentation by a German researcher in 1952, he realized that the Sears-Haack body had to apply to the entire aircraft. This meant that the fuselage needed to be made considerably skinnier where the wings met it, so that the cross-section of the entire aircraft matched the Sears-Haack body, not just the fuselage itself.
Several other attempts to reduce wave drag have been introduced over the years, but have not become common. The supercritical airfoil is a new wing design that results in resonable low speed lift like a normal planform, but has a profile considerably closer to that of the von Kármán ogive. Although the design has been extensively tested, it has not been used, at least in a "pure" form, on any operational designs.