The outer housing that contains and protects a watch's movement, dial, and crystal. Every other structural element, the bezel, lugs, crown tube, and caseback, is either part of or attached to the case.
The case is the structural shell of a watch, machined or cast from metal (commonly steel, gold, or titanium, and increasingly ceramic and ceramic composites) to house the movement while protecting it from dust, moisture, and impact. Every other visible structural feature, the bezel, lugs, crown tube, crystal seat, and caseback, is either machined as part of the case itself or attached directly to it.
Cases are typically built in either a two-piece construction, with a separate caseback that screws or snaps onto the main case body, or a monobloc construction, machined as a single piece with the movement loaded in from the front. The case's material, thickness, and geometry, particularly lug-to-lug distance, affect a watch's water resistance, weight, and how it actually wears on the wrist, arguably as much as the movement inside it.
Stainless steel is by far the most common, with gold, titanium, and increasingly ceramic and ceramic composites used at higher price points or for specific properties like scratch resistance or reduced weight.
A two-piece case has a separate caseback that screws or snaps onto the main body, allowing the movement to be accessed from the back. A monobloc case is machined as a single piece, with the movement loaded in from the front through the crystal opening instead.
Significantly. Lug-to-lug distance, case thickness, and lug design all affect fit and comfort at least as much as diameter alone, which is why two watches with the same diameter can wear very differently on the wrist.