Which is proper when referring to a family named Wilson the Wilson's or the Wilsons?

If you are referring to the family, they are "the Wilsons" (plural).

If you are referring to something belonging to the family, you would use an apostrophe that shows ownership, e.g. "the Wilsons' car".

Note that the apostrophe follows the "s" since you wish to show ownership by the entire family or more than one member of the family. If, however, you wish to show ownership by just one Wilson, an example might be "Sam Wilson's car" in which you have the singular "Wilson" followed by an apostrophe and then the "s".