It was told to me that according to the Torah, a married Jewish man is not considered to violate the commandment against adultery, if he has relations with an unmarried Jewish woman or a divorced Jewish woman. Is this correct? If so, what is the basis for this in the Bible. 

Shalom!

Neither a man nor a woman should not have any promiscuous relations outside of marriage - (source: "Code of Jewish Law - Rma" E.H. 26:1; based on Deuteronomy 23:18).

If the woman is married, it is considered a more severe transgression - (ibid 22:22). This is a technicality related to the fact that according to the Torah, a man may have more than one wife, while a woman has one husband. This is related to the idea that a woman builds a home; a woman cannot build 2 homes.