6 by is short for 6x9 or 6x12 or any other of a specific set of ellipsoidal lighting intstruments used in theater
The "6" in 6 by refers to the diamater of the lenses in the unit measured in inches. The second number refers to the focal length of the lenses. This determines how far the beam of light will spread over a given distance. The further away a lighting instrument is from the object being lit, the narrower the beam will need to be to cover the same area.
For example, a 6x9 will cast a circle of light about 13 feet across from 20 feet away, while a 6x12 will will cast a circle about 9 1/2 feet across from the same distance. This information is very important when designing a show, as it tells you what instruments to use from each location above the stage to get a desired effect.
Most modern elliposidals (such as the Source Four) are measured in degrees of beam spread rather than by the lens size.