An oligarchy is a system of government where a small number of individuals government a political entity. These individuals can be elected, but normally they are unelected officials who have been given the rights to lead through authoritarian methods.
Democracies are a system of government whereby systems are put into place to allow individuals to vote for representatives or to vote directly on issues related to their political governance.
Republics are democracies in that they involve elections and representation, however, republics institute certain protections for individual rights and liberties usually in a constitution or founding document to prevent the majority from controlling political decisions.