Electors in an Electoral College?

Okay. . .so. I am confused. We have this system of an electoral college, save a modified version in Maine and Nebraska. And they are winner-take all style, meaning that the person who wins the most electorates wins the whole states elctoratal votes, which equivalate to the number of Reps in the House + 2 (for the number of Senators). For example, VA has 2 Senators + 11 reps which would equal 13 electoral votes. And we voters vote in our electoral districts or whatnot. But then each state also has two electors selected from the political parties? And they cast a vote for President and VP? I am so confused with how these "electors" come into play. How does this vite work? Can they vote against the voters in the State's district? And are they only chosen from Republican and Democratic parties? What about 3rd parties, like Tea Party people or Constitutionalists? How does our vote for President correlate with an elector?