In the Quake series, Circle-Strafing is running in a circle around the opponent to continuously dodge their attacks. Generally the player will also be firing upon their target whilst using this technique, Circle-Strafing requires the player to be facing their opponent at all times to properly orbit around their foe as well as return fire. It's particularly effective with rapid-fire weapons, such as the Machine Gun or Nailgun. It is done by pressing one of the strafe keys, the forward key, and the turn key.
In Single Player, Circle-Strafing is very effective against most enemies in a close-quarter situation as it allows the player to stay out of range of the enemies' attacks. Note that Circle Strafing is not intended to be used against fast-moving opponents, such as the Fiend, since the enemies' movement will disrupt the circle. Note that the player can potentially Circle Strafe around multiple slow enemies if they can be led to the middle of a room. It is advisable to get enemies as far away from obstacles as possible to prevent accidental collisions whilst Circle Strafing.
A very convenient strategy to use with enemies with homing weapons, such as Vores, is to Circle-Strafe around an enemy whilst being pursued by the projectile. The player's proximity to the Enemy will often result in the projectile making contact with the opponent, whilst the constant movement of the Circle-Strafe will result in the player being safe from harm if done properly.
In Multiplayer, Circle-Strafing is extremely effective, as it allows you to take shots at a target while dodging his. Weapons such as the Quake 3 Shotgun and Lightning Gun become very powerful with Circle-Strafing, as they allow the user to focus on the target while allowing them to easily dodge enemy attacks. For maximum effect, use it in a large room. Small rooms can be Circle-Strafed in, but you have to be aware of your surroundings, or you'll suddenly hit something and stop, allowing a target to get a free shot in.
Be careful, because enemies will be trying to circle-strafe you as well! Practice often, as Circle-Strafing takes time to master.