The video above shows a couple of different methods for cutting and sealing your nylon abseiling or climbing rope. For those who can't be bothered watching, or want to know if it's worth watching, the methods are summarised briefly below.
Using a knife that you don't care about - a blunt one is fine - heat it up in a flame until red hot. Gently use the hot knife to cut through the rope. It may take a few goes of heating it up for the knife to cut through the rope completely. You can then use the side of the knife to flatten out any bits of the end of the rope that are sticking out ("mushrooming"). This will reduce the chance of the end of the rope getting stuck, in cracks for example.
You'll need a sharp knife for this one, as well as some strapping tape. Masking tape or duct tape will do fine. Wrap the spot you want to cut using a piece of tape. Cut through the rope in the middle of the tape using the knife. Heat the end of the rope up in a flame until it lights. Hold the rope upright, with the flame on top. Allow to burn a little, then blow it out. The melting nylon should form a round cap on the end of the rope. Once cool, carefully unpeel the tape.