A complex disease (or condition), when discussed in the context of genetics, reflects a disorder that results from the contributions of multiple genomic variants and genes in conjunction with significant influences of the physical and social environment. For this reason, complex diseases are also called multifactorial diseases. This stands in contrast to a “simple” genetic disease that is more directly caused by mutations in a single gene. Common examples of complex genetic diseases include heart disease, diabetes, and cancer.
In a way, it's sort of funny that any disease would be called not complex, so this is one of those terms that initially seems a little odd, but in our own parlance--and geneticists have their own way of thinking about things--complex disease really is supposed to conjure up in your mind that this is not a simple Mendelian single-gene disorder. It's messier than that. A complex disease would have many genes involved, often significant environmental contributions involved. You might also say it's polygenic, another word that says multiple genes contributing. We're talking about diseases like diabetes, or the common cancers, or heart disease, where you don't expect it's going to be as simple as one glitch in the genome causing the condition.