What is the relationship name of my wife's brother to me?
He is your brother-in-law.
There has already been a similar question posted here to which the answer was "sister-in-law", so I'll finish the list here:
Your spouse's brother or your sister's husband is your brother-in-law.
Your spouse's sister or your brother's wife is your sister-in-law.
Your spouse's father is your father-in-law.
Your spouse's mother is your mother-in-law.
Your daughter's husband is your son-in-law.
Your son's wife is your daughter-in-law.
Where applicable (and where disowning has not occurred), you child's same-sex life partner under a legal bond (whether marriage or civil union or what have you) or a bond you recognise even when the law does not would be either a son-in-law or daughter-in-law, as applicable.