|| || ... is the logical OR function. It evaluates its arguments in order, giving True immediately if any of them are True, and False if they are all False.
Or[, , ... ] can be input in StandardForm and InputForm as ... . The character can be entered as ||, or or \[Or].
Or evaluates its arguments in a non-standard way (see Section A.4.2).
Or gives symbolic results when necessary, removing initial arguments that are False.
See The Mathematica Book: Section 1.5.6.
See also: Xor, LogicalExpand, BitOr, Nor.