When in battle, Cyclops wears his visor for channelling his Optic Energy beams.
Cyclops uses the visor to control his optic beams. Without the visor, he can't open his eyes without emitting a destructive, optic blast.
In the second film, the visor has a more sleek, slim look and no longer covers his ears. This has no other obvious effect other than looking more compact and streamlined and he still maintains the same amount of control over the beams.
When trying to fit in and during the time he isn't fighting as an X-Man, he wears sunglasses made of a similar material to the visor that prevents his optic blasts from destroying everything in sight. These sunglasses also received an upgrade following an altercation in the timeline.
In the final scene in X-Men: Apocalypse, Scott's visor is silver and slim look like a cross between the visors in the first and second film, as well as resembling the visor Cyclops wears in Joss Whedon's Astonishing X-Men comics. In a deleted scene it's revealed that Hank McCoy built the visor for Scott, once wearing it he jokingly mentions how he hopes nobody calls him "Cyclops".