"Nature versus nurture" is a psychology term related to whether heredity or the environment most impacts human psychological development (behavior, habits, intelligence, personality, sexuality, aggressive tendencies, and so on). It's obvious that you share your parents' DNA in the physical sense — you might have long legs like your father and blue eyes like your mother. But where did you get your love of reading poetry, your quick wit, or your natural athletic abilities? That's what the nature versus nurture debate tackles.
Some scientists think that your personality is based on genetic predispositions (nature). Other scientists think the way you act stems from life experience, the way you were taught, and the environment in which you grew up (nurture).
As of now, no one really knows which side wins. Is Kate Hudson an actress because she and her mother, Goldie Hawn, share a creative "gene" that makes them want to perform? Or is Kate Hudson an actress because she grew up in a house with Goldie Hawn, where she learned to love the same things as her mother?
There are plenty of stories about twins, separated at birth, who reunite as adults to discover that they use the same brand of toothpaste or gave the same name to their children. This suggests that genes play at least some role in personality development. But other scientists are taking this further and into ethical questions — what if serial killers were predisposed to commit their crimes? Perhaps the greatest current nature versus nurture debate is over the existence (or not) of a "gay gene," which would suggest that homosexuality is no more a choice than eye color. Obviously, some of these questions reach to the core of some people's morals, values, and religious teachings.
What's most likely is that both nature and nurture play some role in how we all turn out. But no one really knows with certainty. The true answer as to which side tips the scale remains to be seen.