With further elaboration on Ignacia's answer:
A gyroscope itself is generally an analog device in the sense that in reality, it will have a continous output. This output is digitized using some form of Analog to Digital A/D converter where the analog output is converted to a digital number. The output from the gyroscope is a measurement of angular speed, generally millidegrees/sec.
The number read by your Arduino from the gyroscope will be a digital version of your current angular speed at the time of measurement. For example, if you're getting an 8bit number back equal to 121, this in binary is equivalent to 01111001, where the '1' furthest to the right is the least significant bit.
The term "lsb per degree per second" means how many bits change with your angular speed. For example, with a gain of 1 (meaning 1 degree/second = 1 bit), an angular speed of 90 degrees/sec will produce a read value of 90 or 01011010. Therefore, 1 bit changes per degree/second. Increasing the essential "gain" (which will most likely be referred to as "dps" will change how many lsb's change per degree/second.