With relevance to Starcraft, micromanagement is the art of finely controlling your units as to get the most effectiveness out of them. This can range from anything as simple as pulling back the Carrier that your enemy's Wraiths are attacking to make them move closer to your Observer, to laying a Psionic Storm on your opponent's worker line where it will do the most damage to making sure your Siege Tanks are properly spaced out to be able to lay out some Arclite Smackdown. Micromanagement was first brought into the game as an art form by the player named Zileas, who perfected the Reaver Pop to great effectiveness during the Blizzard-hosted Starcraft Ladder finals of 1998 and popularized the concept of trying to preserve your units that they could fight back later.
The three races of Starcraft have different inherent levels of micromanagement to deal with. Zerg players usually use little to no micro, as they are more concerned (and rightly so) with map control and macromanagement. Protoss units are more micro-friendly, especially with their powerful spellcaster units, but Terran units are best utilized by a strong micro-player because their strongest unit- the Siege Tank- requires so much clicking to use, but ends up being so powerful.
As with any part of the game, though, the best players demonstrate a good combination of macro
skills- and because of it, are capable of beating a player who relies too much on manuvering his Dragoon
so that it takes the least possible damage.