It hasn’t happened since 1824 so you can be excused for not knowing.
In that election (4 candidates) Jackson won the most electoral votes but didn’t get a majority so it went to the House, which elected 2nd place finisher JQ Adams. Jackson and Adams both had John C Calhoun as their running mate so he did get a majority of electoral votes for VP.
The Senate had to elect the VP just once, in 1836 democrat Van Buren was elected President but some of his electors refused to vote for his running mate, Richard Johnson, who fell just short of a majority. The Senate elected Johnson over the Whig runner-up on a near party line vote.
This was talked about a little bit during the last election cause some people were hoping McMullin would carry Utah and that neither Trump nor Clinton would get 270. Democrats and enough RINOs in the House could have then possibly elected McMullin.
Some additional clarification, as this subject is a bit more complicated than the way that Impy and DJ described it (although their examples are correct).
If no presidential candidate receives a majority of electoral votes (currently 270), the 12th Amendment provides that the House gets to elect the president, choosing “from the persons having the highest numbers not exceeding three on the list of those voted for as President.” There are two ways to read this: As stating that the House may not consider more than three candidates, or that the House may not consider more candidates than those with the third highest number of elections. I’ve thought about this for over 20 years now, and my conclusion is that the most logical reading would be that if, say, Trump has 266 EVs, Hillary has 264 EVs, McMullin has 4 EVs and Stein has 4 EVs, that the House could elect any one of those four (because they are the persons that received the top three “numbers” on the list). So, if I’m correct, the House wouldn’t be limited to three persons if there’s a tie for third; furthermore, if the first-place finisher got 238 EVs and there’s a three-way tie for second with 100 EVs each, the House could choose from among four candidates instead of being limited to just one choice.
A crucial point is that, when the House votes to elect the president of the United States, the vote is held by state delegation, not individually. So if California’s 55 Representatives cast 30 votes for Hillary, 20 votes for Trump, 3 votes for Stein and 2 votes for McMullin, Hillary would receive 1 vote in the contingent election; meanwhile, if Alaska’s sole Representative cast his vote for Trump, Trump would receive 1 vote in the contingent election.
Moreover, if no candidate gets a plurality (or, if the House votes to require a majority, a majority) of a state’s delegation, no vote would be cast by such delegation. So if New Mexico’s Representatives cast one vote for Trump, one for Hillary and one for Stein, no one would receive the vote of the NM delegation. Since *a majority* of delegations are required for the House to elect a president, these ties increase the probability that none of the candidates would be elected president in the first round of voting.
If no one receives the vote of 26 state delegations, a second vote would need to be held, with Representatives being able to change their votes, until someone is elected president. (If no one is elected president by January 20, the new VP would be sworn in as Acting President and serve in such capacity until the House finally elects a president; as Acting President, the VP can make all sorts of appointments and sign or veto bills, but would *not* be entitled to nominate a new VP under the 25th Amendment, for the simple reason that *there is no vacancy in the vice presidency* (he is the VP, and merely is acting as president until the House elects one).
Which may lead you to ask, how did the VP get elected? If no candidate received 270 VP EVs, the 12th Amendment provides that the Senate, by a vote of “a majority of the whole number” of Senators (i.e., 51 Senators, with the lame-duck VP *not* being able to break a tie), would elect the VP “from the two highest numbers on the list.” The language here differs from that used in the case of contingent elections in the House, so there’s no danger of someone claiming that if VP candidate X got 180 EVs and VP candidates Y and Z each got 179 EVs that the Senate only could elect X. However, there’s a different potential problem with this phrasing: Wooden literalism might lead someone to argue that if X and Y each got 268 EVs and Z got 2 EVs that the Senate could consider Z because 2 is the “second highest number” (with 268 being the highest). However, I would hope that judges wouldn’t be so mathematically clueless as to fail to realize that the two highest numbers in such case are 268 and 268.
As for what happens if the House fails to elect a president *and* the Senate fails to elect a VP, that would take a much longer post to explain. Cliff’s Notes version: Both the new Speaker and the holdover Secretary of State would claim to become Acting President on January 20, and we’d be in the midst of a genuine constitutional crisis. If you’re interested in this issue, as well as other potential constitutional crises, I highly recommend “Constitutional Cliffhangers” by Brian C. Kalt: https://www.barnesandnoble.com/w/constitutional-cliffhangers-brian-c-kalt/1102887882?ean=9780300234305