If you have the basic knowledge of the terms that make up these words, it gives us idea about what these words might most likely mean. "Poly" means many, "bi" means two, and "gamy" means marriage. Hence, we have a clue that polygamy can be related to "many marriages", while bigamy can be related to "two marriages". But normally, bigamy is actually an offence in the court of law where a legally married person gets involved in another gets involved in another marital affair.
Polygamy, in contrast, simply refers to a phenomenon when one (a man) is married to many wives. So the different between these is not basically in the number of spouses or marriages. While bigamy is considered a crime, polygamy is considered a religious practice (among the Muslims especially). Bigamy has to do with the two spouses not knowing themselves and living in different households, while in polygamy, the spouses know themselves and are most likely living in the same household.