A microbiome is the genetic information of the 100 trillion microbes which help a complex organism carry out its life functions. A microbiome is unique to each individual within a species.
Both a microbiome and genome may "define" an individual on a purely physical level. The microbiome differs from the genome in that the microbial composition within a complex organism changes drastically over time. Typically a genome changes very little if at all.
Microbes attach to an organism in a high concentration at a number of different spots such as the belly button, nose, ears, mouth and GI tract. By sampling microbes from these spots, an individual's microbiome may be mapped.
For mammals born with an umbilical cord, the microbes within the navel are passed down from the mother. For many people these microbes remain relatively unchanged from birth.
Currently there are at least three different projects seeking to map out the human microbiome on a large scale. The first (and most likely to succeed) is the National Institute for Health's Human Microbiome Project. The NIH is requesting that only "healthy" individuals participate in their sampling.
The other two are uBiome and American Gut Gut and are open to people of all levels of health. Both of these projects eschew formal scientific funding in favor of the crowdsourcing platform indiegogo. uBiome is the smaller project in scale and as of this writing has reached a higher percentage of its goal for funding. Both projects seek to entice individuals with a greater knowledge of the forces operating upon their lives which they do not understand. The American Gut project has a strong focus on gut flora.
In an age where many health-conscious people regularly consume probiotic capsules or make fermented foods such as yoghurt or saukraut a staple of their diets, it seems natural for a sampling and mapping of microbiomes to be of interest. How representational that microbiome is of the average person is another question altogether.
Human Microbiome Project
For information on the effect antibiotics have on the microbiome, check out this blog post.