As Article 2 of the Constitution says, the President and Vice-President are chosen by electors, and the electors can be chosen however the state legislature wants. Since 1860, with one exception, every state has had the people elect electors who (sometimes unofficially; sometimes officially) pledge to cast their vote for a particular President and Vice-President. Once the elector is chosen, the voters have no further official influence over who he votes for. So, when you cast your ballot, you were actually voting for a slate of electors who'd promised to cast their separate Presidential and Vice-Presidential votes in a specific way.

Originally, a number of states did have the legislature choose electors, sometimes changing the method from election to election. For example, in 1800, the Federalists in Pennsylvania's upper house insisted each house choose electors separately to block the lower house's Republican majority from sweeping the ticket. In the early 1800's, the rising trend of Jacksonian democracy led to every state giving the choice to voters except South Carolina. After the Civil War changed that (to make a long story short) the one last exception was Colorado in 1876, which was admitted as a state too late in the year to organize a popular vote.