Nearly everyone, not quite the majority (6)
I believe the answer is:
almost
'nearly' is the definition.
('almost' can be a synonym of 'nearly')
'everyone not quite the majority' is the wordplay.
'everyone' becomes 'all' (I've seen this before).
'not quite' means to remove the last letter.
'the majority' becomes 'most' (I've seen this before).
'all' with its last letter taken away is 'al'.
'al'+'most'='ALMOST'
(Other definitions for almost that I've seen before include "Nearly, not quite" , "practically" , "Not entirely" , "As good as" , "Slightly short of" .)