The "cycling" process is where the benificial bacteria which do the important job of converting Ammonia through to Nitrate are breeding up so that they can handle the "load". The bacteria are mostly in the growbeds as that provides the largest surface area for them to grow on.
If you have a test kit (you should) which tests Ammonia, Nitrite and Nitrate, and you start a system going with a few fish in it, you will see Ammonia rise over a few days/weeks depending on the size of the system. Then as that form of bacteria breed up the Ammonia drops off and the Nitrite starts to increase. When the Nitrite starts to drop off the Nitrate will increase. When you have Ammonia and Nitrite readings of zero, then your system is cycled for the "load".
I say it's cycled for the "load" because if all of a sudden you dump in another 100 fish, there won't be enough bacteria to handle all that extra input, and the system will take time to cycle up to handle that load.
Important to note that Ammonia is poisonous to fish, as is Nitrite (but less so). So if you've got a prize crop that you want to grow, you want to start with a cycled system.
You can shorten the cycle time by innoculating the system with the bacteria from a working system (or fish pond or fish tank).