Legally, bigamy is when a man or woman is married to two (or more) partners simultaneously. This practice is illegal in the western world and generally exists with at least one of the partners being unaware of the other's existence.
For instance a man might travel a great deal for his work without his wife and meets another woman that he forms a relationship with and subsequently marries her also. He is now juggling two wives, two homes and maybe two families. This is a very tricky relationship; expensive, time-consuming and legally risky. When caught, the legal system considers this fraud and prosecutes it as a felony. Unfortunately, this scheme is used by some criminals to defraud partners--they court and marry a partner, take him or her for everything they have, then leave and repeat the scheme on another innocent partner.