The cycle is basically you growing good bacteria. It takes 2-6 weeks for these bacteria to be established and plenty of time to convert ammonia to less toxic nitrate. Ammonia/nitrites are a waste product which are produced by rotting food or fish waste, without the good bacteria the ammonia will keep building up until the fish eventually die.
The bacteria first deals with the ammonia, this is converted into nitrites which are still poisonous. The bacteria continues and this is then converted into something much less toxic, nitrates. If you use a test kit on a cycled tank the healthy reading should be 0 ammonia, 0 nitrite and under 20ppm of nitrates. When the nitrates start to raise over 40ppm then this is bad and will eventually start to kill off your fish too
What you can do to start the cycle is either feed your tank with fish food once every 12 hours, add a dead shrimp or piece of raw fish, also if you can hands on old filter media or substrate this will speed up your cycle too
There are plenty of articles on the internet about cycling a tank, but it always helps to ask here and get a broken down version