In my layman understanding: Given a set, a quorum is some method to choosing a sub set, such that any two such sub sets will always have at least one overlapping member.
Majority is one quorum algorithm - given a set [A,B,C], the majorities are: [A,B,C], [A,B], [A,C] and [B,C]. Any two of those sets will have at least one member overlapping.
However, majority is somewhat wasteful, because the latency of these quorum-based algorithms are almost always bound by the slowest member of the quorum - the more machines you need to wait for, the more likely one of them will be outlier-slow.
You'd potentially be better off choosing a quorum algorithm that requires less than a majority - because that'd mean, in the best case, fewer responses to wait for, lowering the probability that one of those members will be very slow. There are drawbacks to this - it makes fault tolerance and provisioning harder to calculate - but it's got some cool potential benefits.
Some cool ones to explore here: https://pdfs.semanticscholar.org/a243/7f18205414f6398b29c4f8...