Keep it clean and dry. Some people advise using cord spirit, but I just cleaned it with some gauze and cool boiled water in upward strokes. Don't bathe your baby or submerge the cord stump in water until it has dried up and fallen off. Don't pull or tug at it and let it dry on its own. This will take about a week or two. My girl's stump fell off in 4 days, which was considered pretty fast. http://sg.theasianparent.com/bathing_your_baby_melissa_lau/
Like what mummy Jamie says, keep it dry and clean. Never use tap/soapy water as the bacteria in it might cause an infection. My son's cord dropped off in slightly less than a week.
thanks