Far East (in contrast to Middle East) refers to countries/territories of East Asia e.g. China, Japan, Koreas, Hong Kong, Taiwan etc. However, South-east Asia region is usually added to be considered as such Far East too, some SE Asia states are Malaysia, Singapore, Thailand, Indonesia. Hence Far East means East Asia (sometimes plus SE Asia).
Far East seems to be a term coined and used more frequently by the British/Europeans and had its origin since their colonial era in East Asia. Middle East (or West Asia) in contrast (at least the modern meaning of it) is used frequently by the Americans.
Far East is an inexact term often used for East Asia and Southeast Asia combined, sometimes including also the easternmost territories of Russia, i.e., the Russian Far East, and the western Pacific Ocean region.
Far East is sometimes used synonymously with East Asia, which may be defined in geographic or cultural terms to include historical China (excluding Tibet and Xinjiang), Japan, Korea, and Vietnam.
Far East in this sense is comparable to terms such as the Orient, which means East; the Eastern world; or simply the East.