About the electoral college method of electing presidents

On the day after the election (the Tuesday immediately after the first Monday in November), all the votes have been counted, everything is settled, we have a new president!
Umm, not exactly.  The actual vote for president occurs in December, specifically on "the Monday following second Wednesday in December" when the "electors" meet.  The result of that vote is presented to Congress (some of whom may have just been elected) on January 6, two weeks before Inauguration Day, and a president is declared - if someone received at least 270 votes.

"Every vote counts", right?

For local elections, they certainly do.
For presidential elections, in 48 states, every vote counts right up to the time a winner is declared and all the votes are transferred to the winner.  Maine and Nebraska attempt to apportion electors matching the popular vote.
Each political party in each state provides a list of "electors" - there are a minimum of three in a state, up to fifty-five(55) in California.  Whichever candidate wins the state vote, their party's electors then represent the entire state.  They are not generally required by law to cast their electoral vote for the same candidate that won, but they usually do so.

What if no candidate gets the 270 electoral votes?

If no candidates achieves a clear majority of electoral votes, then:

• For President: The House of Representatives selects the president from the three (3) candidates receiving the most votes.  Each state gets one vote, regardless of population.  A simple majority of twenty-six (26) votes determines the President.
  
• For Vice-President: The Senate selects from the two (2) candidates receiving the most votes. Each state gets two votes.  A simple majority of fifty-one (51) votes determines the Vice-President.

It is possible the President and Vice-President could then be from different parties.
If the House fails to make a decision, the selected Vice-President becomes the acting President until they do so.  If the Senate has failed to make a decision, the current Speaker of the House becomes acting President.

Is it possible that a third-party candidate could win?

Is it possible that a major party candidate could be:

• rated unfavorable by a majority of voters?
  
• someone under FBI investigation for a possible felony?
  
• someone with zero government experience who insults large groups of people?
  
• someone named in a RICO (racketeering) lawsuit for over 5 years?
  
• a democratic socialist running for leadership of a republic?

Or that polls show, compared to the two major candidates,:

• 5% preferred the dead gorilla Harambe, which had lower unfavorable ratings than both?
  
• 13% of voters would prefer a giant meteor hitting the earth?

So, in 2016, the year that nothing is normal - a third-party win would not seem out of line.

But there's no way a third party can get 50%

That is likely true - and irrelevant.  Thanks to the winner-take-all rules in 48 states, a candidate only has to win the most votes (a plurality), not the majority.  So, in a tight three-way race, someone could win a state with 34% of the popular vote.  In a tight four-way race, they would only need 26%.  2016 is an election with four national candidates, so someone winning with 30% of the vote is possible.

Here is one assesment of how someone could become president by winning just one state.

Local elections - more important than presidential?

Electing a president is important, but that's not who fixes your roads.  If you don't know who your county/borough/parish commissioner is, or who your city's Mayor is, you should find out - those folks affect your daily life.
As government in the 2000's has shown, national governance is mostly bark and little bite...

How government really works

Find an old (1980's) British television show named "Yes, Minister" which brilliantly shows just how much elected national officials matter.  Seriously.  Watch it.  But still VOTE!