x && y executes x, then only executes y if x succeeds (i.e., has an exit status of 0)

x || y executes x, then only executes y if x fails (i.e., has a non-zero exit status)

The output of x is irrelevant.

One place they do not differ is in precedence. Unlike Boolean operators in other languages, && and || have equal precedence, so something like

a || b && c

is parsed and evaluated the same as

{ a || b; } && c

rather than

a || { b && c; }