In competitive Brood War "Random" refers to players who do not select a race before the game starts. Players who select a race for a specific match up are called Racepickers.
Playing random was more common in the earliest days of competitive Brood War. Especially in the official Battle.net Ladder and in tournaments not selecting a race had several advantages. For one, the opponent was forced to play rather defensive as he couldn't anticipate which race he would face. At the same time, the random player could decide which kind of gamestyle he would force onto his enemy - ranging from very aggressive to a somewhat standard opening.
However, with increasing knowledge about the game, as well as maps featuring longer rush distances choosing random became less powerful. The player to play random had to master all nine possible match ups, whereas his opponent only needed to understand his own race. The Korean professionals therefore focussed on one race or at least on specific match ups (racepicking). Additionally, the inclusion of rules forbidding to play random in tournaments played a role in the decline of random players.
Choosing random in professional tournaments and leagues was mostly restricted to 2v2 team play. At times the combination of two Zergs within a team was forbidden, as the combination allowed for rather strong strategies. To circumvent this ruling one player sometimes picked Random, while his ally chose Zerg. This way a 33% chance was maintained to spawn as Zerg/Zerg team without violating the rules.
Notable Random Players
In the earliest days of StarCraft Brood War mostly foreigners were somewhat successful as random players. Among them were the German Infinity and the Norwegian Eriador, both of which were featured in the Pimpest Plays. The by far most successful random player in the earliest days of Brood War was the Chinese Pj, who already won several minor titles before being forced to play Protoss in SK Telecom T1. In the later years the Canadian Testie was known to sometimes pick Random in bigger tournaments. Most other random players missed to win bigger events. However players like Kashu and Napoleon achieved to hold decent records in high level clan leagues. For training purposes some foreigners like TechnicS and Mondragon were known to switch to random.
After the Wings of Liberty Beta more random players emerged for a short while. Aforementioned Kashu and TechnicS, as well as Michael, NeMu and funNy showed decent results in small scaled tournaments using random.
In minor non-money tournaments the "no random" rule was temporarily turned upside down. For instance the Xsplit Random Invitational forced all participants to use random instead of their respective main races. Random as race was forced upon professional KeSPA players for show matches like the Battle.net Attacks. In the international scene the same handicaps were used for copies of the Korean version.