Temporary orbit used during the launch of a spacecraft
A parking orbit is a temporary orbit used during the launch of a spacecraft. A launch vehicle follows a trajectory to the parking orbit, then coasts for a while, then engines fire again to enter the final desired trajectory. 
An alternative trajectory that is used on some missions is "direct injection", where the rocket fires continuously (except during staging) until its fuel is exhausted, ending with the payload on the final trajectory. This technique was first used by the Soviet Venera 1 mission to Venus in 1961.
Reasons for use.
Geostationary spacecraft.
Geostationary spacecraft require an orbit in the plane of the equator. Getting there requires a geostationary transfer orbit with an apogee directly above the equator. Unless the launch site itself is quite close to the equator, it requires an impractically large amount of fuel to launch a spacecraft directly into such an orbit. Instead, the craft is placed with an upper stage in an inclined parking orbit. When the craft crosses the equator, the upper stage is fired to raise the spacecraft's apogee to geostationary altitude (and often reduce the inclination of the transfer orbit, as well). Finally, a circularization burn is required to raise the perigee to the same altitude and remove any remaining inclination.
Translunar or interplanetary spacecraft.
In order to reach the Moon or a planet at a desired time, the spacecraft must be launched within a limited range of times known as a launch window. Using a preliminary parking orbit before final injection can widen this window from seconds or minutes, to several hours. For the Apollo program's crewed lunar missions, a parking orbit allowed time for spacecraft checkout while still close to home, before committing to the lunar trip.
Design challenges.
The use of a parking orbit can lead to a number of technical challenges. For example, during the development Centaur upper stage, the following problems were noted and had to be addressed: 
The Centaur and Agena families of upper stages were designed for restarts and have often been used in missions using parking orbits. The last Agena flew in 1987, but Centaur is still in production. The Briz-M is also capable of coasts and restarts, and often performs the same role for Russian rockets.
References.
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