Generally it is considered inappropriate to reject a handshake, and in most social circles it is expected that the one with higher social status will initiate it. To offer the left hand instead of the right was once considered an insult.
Handshake is originally a Western European tradition, brought into practice in its present form by English Quakers in 17th century as a more egalitarian and simpler alternative to the complex greeting etiquette of the contemporary higher social classes.
Today handshake is known and practiced throughout the world, though in many cultures alternative customs for greeting, such as bowing or cheek kissing are still preferred.
Handshakes are a major route for the transmission of common cold viruses.
See also: holding hands, Handshake Man