Nearly 2.9 million Americans voted for Hillary Clinton than for Donald Trump (see e.g. The Cook Political Report's final tally), but back in late November Trump tweeted that he actually won the popular vote for president "if you deduct the millions of people who voted illegally."
He offered no evidence for this, and voting officials of both parties have rejected the claim as clearly false, as have independent investigators. A January 25 report on CNN observed that
A number of studies have found no evidence of widespread voter fraud.
The Truth About Voter Fraud, a report written by experts at The Brennan Center for Justice, found voter fraud rates were between 0.00004% and 0.0009%.
"Given this tiny incident rate for voter impersonation fraud, it is more likely that an American will be struck by lightning than that he will impersonate another voter at the polls," reads the report.
Other news reports have accurately described Trump's claim as "false" or "debunked," but I don't think it's been made clear enough that his claim simply doesn't even make sense from the standpoint of basic arithmetic. That is, there's no plausible way it could be anywhere close to true.
See, for example, the January 26 article by Politico's Chris Ashby, who calculates the wildly implausible magnitude of fraud that would have been required.
And we should recall that Trump isn't just asserting massive voter fraud. He's more specifically said that the votes in question came from 3 to 5 million illegal immigrants. According to multiple sources, he repeated this claim during a January 23 meeting with members of Congress. Quoting Fox News, "President Donald Trump claimed at a meeting with congressional leaders Monday that he lost the popular vote to Hillary Clinton in last November's election because between three million and five million 'illegals' cast ballots, multiple sources told Fox News." Or as the Associated Press reported it, "President Donald Trump asserted in a private meeting with congressional leaders Monday night that he would have won the popular vote in the 2016 election if 3 million to 5 million immigrants living in the country illegally hadn’t voted."
Yesterday Kevin Drum noted in passing, "There are about 20 million noncitizens in the US. Trump is therefore saying that 15-25 percent of all noncitizens voted. This is fantastically beyond anything even remotely plausible." (That's just one point he makes; read the whole piece. He also points out that a 2014 study cited by Trump himself actually suggests that very few if any noncitizens try to vote.)
And Trump isn't just claiming that noncitizens voted but specifically that 3-5 million illegal immigrants voted. The Department of Homeland Security (link to PDF) and the Pew Research Center (link) agree that there are about 11 million illegal immigrants in this country. More than two million of them are children, so there are at most 9 million old enough to be mistaken for an eligible voter. Trump's 3-5 million would be (at least) 33 to 56 percent of them.
Incidentally, according to Trump, there were no illegal votes cast for him. Or so he told David Muir of ABC News in an interview that aired Wednesday night. But a January 26 piece by Aaron Blake for The Washington Post points out that of four known cases of illegal votes in the 2016 election, one was cast by some claiming to work for the Trump campaign and another by someone who claimed to have voted for Trump. A third was registered as a Republican. (The original report is here.)
Finally, on January 25 Trump tweeted, "I will be asking for a major investigation into VOTER FRAUD, including those registered to vote in two states, ..." As various sources, including an article in yesterday's New York Times, have reported, people registered to vote in more than one state include his daughter Tiffany, his press secretary Sean Spicer, his advisor and son-in-law Jared Kushner, his nominee for Secretary of the Treasury Steven Mnuchin, and (at least until a few days ago) his strategist Steve Bannon.
Being registered in two states is actually very common and typically occurs when someone moves from one state to another. It doesn't mean that the people in question have voted twice.by