a "majority" is any number bigger than 50%, but it usually implies something closer to 80%.
a "slim majority" is 50-60%: it is a majority, but not a strong majority.
It looks like you are talking about an episode of "American Dad" from 2005. At that time, "red states" (states that voted Republican) contained a slim majority of Americans.
FYI, if there are 3 or more groups, and none of them is big enough to be a majority, the largest group is the "plurality".