This is clearly a portmanteau word: microbe + genome. Microbe refers to microscopic organisms - in this case, bacteria; genome refers to the sum total of genetic material in an organism.
There's a community of bacteria living within humans that performs functions essential for human life. Scientific American (Mar-2012) says there's at least 10 times as many bacteria cells as human cells in the human body (but those bacteria are a whole lot smaller, with much, much simpler genome).
So microbiome is the genetic material of the [useful] bacteria in the human body. In such a situation, it differs from human to human, so things like this are usually measured on a sampling basis.
Small bone to pick here. The human genome is generally thought of as referring to the genetic material (23 chromosomes of DNA) housed in the nucleus of each cell. But there's extra DNA not in the nucleus: mitochondrial DNA, used to generate energy, passed on only maternally, and originally passed to humans by bacteria. This DNA is often left off discussion of genomes. In this case, I'd say they'd be including mitochondria for the sake of completeness.
Another small bone to pick. Microbiome is listed in Wikipedia as the sum total of microscopic organisms in a particular environment, and hologenome is used as an idea of co-evolution with microbes, roughly speaking. So the Scientific American article is pushing the envelope a bit.
So even within a scientific community the meaning of words can change over time until/unless locked down.
Further reading (both from Scientific American, as it happens):