hey blipblop, welcome to AA.
The "quick and dirty" version of cycling goes like this. When you put your fish in there, they create waste (ammonia), and when you feed them, any uneaten food sits at the bottom of the tank (ammonia). Ammonia can build up VERY fast and the water will become toxic, and burn their fins and gills.
After some time the correct bacteria will grow to convert the ammonia into nitrite. Nitrite is less toxic than ammonia, however is still very toxic.
Once again, over time, the correct bacteria will grow to conver the nitrite into nitrate. Nitrate is the least toxic of them all, and is the end of the cycle. At this point the only thing you can do is water changes to get the nitrates down.
And there you go, your cycle is complete. Now the trick is you want this all done BEFORE you put fish in, so that they don't have to suffer through the burning ammonia and nitrite. So the most humane way is fishless cycling. Because you can't just setup your tank and let it run for a week and expect it to cycle, there needs to be a source of ammonia to start the process. So you can either add actual straight ammonia, or a common method is putting a fresh shrimp from the grocery store in the tank.
I believe both methods are talked about in the articles section (link at the top of the page). Good luck!
-brent