Traditionally meant, "For the Republic," connotating "One for the people," as opposed to One who sides with an ideological "Empire," and initially had NOTHING to do with Right/Left-wing politics.
Ideologically speaking, Republicans' sworn enemy is not the Democrat, but the Imperial. However, there has been a switch in the labelling, and the party with interests in helping people by means of utilizing a/the Government run by the people (Republic/Republican) is actually a political view held by the Democrats (for example, Welfare, Civil Rights, Health Care, Education, ETC.)
The now-labelled "Republicans" bear a closer resemblance to Imperials (Exploitation, inconsideration, coercement of beliefs, etc.) than the traditional definition of Republicans.
Modern Democrat = Traditional Definition Republican
Modern Repubican = Imperial