How many types of organisms are in your body?

No one knows. It is estimated that there are between 500-1000 different species in the human gut alone. An interesting finding has been that obese people typically have less diversity in the gut microbiome compared with lean adults. In general, less microbial diversity in the gut is associated with more chronic disease.
100 trillion. According to the human microbiome project there are 100 trillion bacteria in or on your body.