Sure thing. Here's a quick breakdown on the important bits:
pH - this is whether your water is acidic (below 7), neutral (7) or alkaline/basic (above 7). Bettas like their water neutral or slightly acidic, but as they have been bred and domesticated they have become more tolerant of different water conditions and can cope with basic water too. :) If water is too acidic or too basic, it can hurt fish by denaturing their proteins, which is changing the shape of the actual protein. This is bad.
Hardness: this is measured in GH (general hardness) and KH (carbonate hardness). Bettas prefer softer water, as do tetras, loaches and cories. Livebearers prefer harder water. I believe that the wrong level of hardness can effect kidney function. My water is ridiculously soft (1 degree of hardness).
Ammonia: ammonia is toxic in high quantities. It is the byproduct of fish waste as it breaks down. Ideally, you want ammonia to be 0, and should not let it get above .25ppm. High levels of ammonia burn a fish's skin and gills.
Nitrite: nitrite is what occurs when little things called nitrifying bacteria settle on surfaces, particularly filters, in your tank, and nom on the ammonia. Essentially, they poop out nitrites. Nitrites are also toxic in high quantities, so again, don't let them get above 0.5ppm.
Nitrate: the third stage of nitrifying bacteria poop. The bacteria essentially eat the nitrites and poop them out as nitrates. Nitrates are still dangerous, but far less so then nitrites or ammonia. Ideally, you want to keep nitrates under 20. Anything above 40 is bad.
Of course, this is a massively simplistic description of these things. :p Essentially, a cycle is when the bacteria have colonised your tank and converted the ammonia to nitrites and then nitrates. :)
I hope this helps provide a starting place for you. :)