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".

It means 'few' used like this.