The answer is 'a whole bunch!'. Wikipedia says there are about 100 Trillion = 1014 of them. You can make a crude estimate, however, without counting all of them. Individual cells can be seen in a microscope and you can measure their size, d, which is about 10 microns.
The volume of a cell is then d3 and its weight is the volume times the density, close to that of water. Putting this all together you get that a single cell has a mass around 10-12 kilograms. Dividing into the weight of a good sized person, 100 Kg , you get the Wiki-number.
p.s. It turns out that there are something like ten times as many bacterial cells in your body as there are cells with your DNA. The bacterial cells are much smaller, so they only account for roughly 1-3% of your weight. See: http://en.wikipedia.org/wiki/Human_microbiome
(published on 09/10/11)
(published on 06/11/12)