What happens if neither presidential candidate reaches 270 electoral votes?

It goes to Congress like it did in 1876.   Congress come Jan 2021 gets to make the call.   All this stuff about Trump filing lawsuits and using the courts is just BS,  it is Congress that gets to make the call if neither has the 270 votes to win.  

Well we've only got two real candidates in the race.  There are other candidates, but they have no chance of winning any states.  So it's incredibly unlikely that someone will not get 270 votes.  The only way that would happen is a 269-269 tie.  To get that you'd have to hit the exact right combination of states.  The most plausible scenario for this would be for Joe Biden to win everything that Clinton won plus Michigan, Pennsylvania, and then the 2nd Congressional district in Maine (Nebraska and Maine award electors based on who wins each Congressional district).  Then it would be a tie. 

The Constitution has a procedure for this.  If no candidate has an outright majority of electors then the election goes to the House of Representatives, which will pick the President from the three candidates who got the most electoral votes.  But, it's not just a simple vote of the House.  Instead, each state delegation votes, and they all have an equal vote. 

Yes.  What happens is that the 50 state delegations in the House of Representatives choose from the four candidates with the most electoral votes (in practices, there are usually only two or three candidates with any electoral votes, so the choice would be between them).   This last happened in 1824.

For that to even be possible, we would need a third party who is actually capable of getting an electoral college win.

The house of representatives would decide. The Senate chooses the vice president.

The electorial college gets to make the decision.