Well... in a nutshell a cycle is just that... a cycle (nitrogen cycle to be exact.)
Your fish excrete ammonia through their urine and poo (and uneaten food, as well will contribute to ammonia developing in your tank). Ammonia is toxic to your fish and can cause burns and aggravate diseases/parasites. During the cycling process... a bacterial colony starts to form (aka "good-guy" bacteria) in the filter and on the gravel and decorations... and converts the ammonia to nitrite. Nitrite is still toxic to a fish. Again, during the cycling process... another colony of good-guy bacteria starts to form and converts the nitrites to much less toxic nitrates. You get rid of those nitrates by doing partial water changes and gravel vacuuming on a weekly basis.
Some conditioners like Prime will help to convert the ammonia to ammonium (which is non-toxic)... and the good-guy bacteria will still eat that up.
Many will do a cycle with their fish, but typically bettas don't handle it well... as they can't handle prolonged exposure to ammonia and nitrites very well. A typical cycle with fish can take 4-6 weeks.
Many will do a "fishless cycle" - whereas you use pure ammonia instead of fish (the ammonia has to be 100% pure ammonia and water - no surfactants, fragrances or other additives.) It takes about as long - but is safe b/c no fish are exposed to the high levels of ammonia and nitrites. I tried a fishless cycle but it stalled 3 weeks into it and being new - probably messed it up somewhere... so I gave up and moved on to... BioSpira.
I personally have used BioSpira (made by Marineland) with great success in cycling my tanks! (And no, I'm not an employee of the company who makes it.) It's basically the "good-guy" bacteria in a package that you add directly to your tank... you don't have to wait for them to establish themselves... they get right to it as soon as you add them and cycling typically is done within a week or two... with very low levels of exposure to ammonia or nitrites, if any. This is a refrigerated product - and is not to be confused with other products like Cycle or Stability.... which are pretty useless IMO and are basically dead bacteria that seldom comes back to life. With teh BioSpira, my tanks were cycled within a week and my bettas did beautifully throughout (although I panicked at one point with an ammonia spike of 0.25ppm)... but I was able to handle it and get it back down to zero with a water change and adding a bit more Prime water conditioner (which remember converts ammonia to ammonium.)
I have a 5 gallon cycled with one betta and 2 - 3 gals with 1 female in each. I have a new male betta currently in a 3 gal, which I will be transfering to a 5 gallon this weekend (and will cycle it again.) I only bought a 3 oz package (on line - shipped overnight with a cold-pak) and I still have loads left. Some say it doesn't last very long, but I know some that have used it up to six months and there was still some live bacteria. I also have heard of some that have not had good experiences with it... but personally speaking - I think it's a wonderful product.
Actually, that reminds me of another method you could use in place of BioSpira - if you know anyone that will give you some of their gravel or filter media from an established and disease-free tank - that would also work. But IMO, it's risky incase there is a hidden disease or problem that you could be infecting your fish with.
Sorry if that was information overload.
