In American nautical jargon, full speed for a ship is generally termed full ahead; this is as fast as the ship can safely go, as fast as it can maintain for a long period, or, in some cases, simply gives the desired fuel efficiency. It is generally one notch up from 'standard speed'. However, in battle or in an emergency, many ships can exceed their safe maximum operating speed for short periods. This is known as 'flank speed' or 'flanking speed'.
Flank speed tends to eat up a lot of fuel, and often pushes an engine past its designed maximum output. This can be quite dangerous when using steam or nuclear engines, and can cause problems even with less explosive fuel types. Flank speed might be used to avoid an enemy formation, escape enemy planes or depth charges, or to race an errant iceberg. Modern military ships may be able to maintain flank speed for long periods of time, with the only downside being high fuel usage, and if the cause is important enough an entire journey may be taken at flank speed.
There are two main theories on where the term came from. One is that flank speed is the burst speed needed to flank an enemy ship prior to firing on them (or perhaps, the burst of speed needed to avoid being flanked); or alternately, the speed that a ship on the outside of a turning maneuver would need to accelerate to in order to maintained its position on the flank of the formation. The other is that in order to achieve maximum speed, some ships would use the smaller maneuvering engines on the side of the ship -- the flank engines -- in addition to the main engines to gain more speed.
Flank speed can involve a significant increase in power. For US nuclear submarines, full speed is 50% of reactor power, while flank speed is 100%. How much of a difference this makes in actual speed will not be limited by the reactor, but rather the specifics of the turbines, which can only throughput so much energy. The difference in full and flank speed depends very much on the specific ship and engine.