What Isthmus said.
For now, Minor leaders do not have a unique mechanic (this will be remedied in 1.4). They are simply leaders that are not as important, and have different/oddball traits.
Emergent leaders start with one trait, nearly always unique to them, designed with their civ in mind. They are strong, but pay for it by starting with just one trait... They can gain the second as you play.
All three can be seen in the scoreboard.
Two Chevrons + Star, in Purple (first line) is Major.
One Chevron + Star, in Blue (second line) is Minor
One Chevron, in Green (third line) is Emergent.
This thread should have more info, as well as specifics on each individual leader; It's become a bit out of date, but is mostly still valid.