A takkanah is a rabbinic enactment: not an interpretation of scripture but a new rule, made by a competent authority for the good order of the community. The two that made Gershom's name concern marriage.
The first forbids a man to be married to more than one woman at a time. Biblical and talmudic law permit polygyny, and Jewish communities in the Islamic world practised it into the modern period. The Rhineland ordinance closed it in Ashkenaz, and the closure held so firmly that within a few generations it was treated as though it had always been the law.
The second forbids a husband to divorce his wife against her will. Under talmudic law a divorce is effected by the husband delivering a bill of divorce, and the wife's consent is not required. The takkanah supplies it. Together the two ordinances gave a Jewish wife in northern Europe a security her contemporaries elsewhere did not have, and Avraham Grossman has argued that this is part of a wider improvement in the position of Ashkenazi women in the same period.
The third forbids opening and reading a letter addressed to another person, an early and often cited rule of correspondence privacy in a society whose commerce ran on letters carried by third parties. The fourth forbids reminding a Jew who was forcibly baptised, and has returned, of what he did under compulsion.