I've broken down each of the three prongs below. Keep in mind that congressional committees are conducting investigations of their own: This post covers only the special counsel investigation. To better understand the others, check out Amber Phillips's guide.
Russian election meddling and possible collusion with the Trump campaign
This is where it all started. James B. Comey, who led the law enforcement investigation until he was fired as FBI director May 9, testified last week before the Senate Intelligence Committee that he has no doubt that Russia attempted to influence the presidential race by hacking the Democratic National Committee and launching cyberattacks on state election systems, among other tactics.
Comey's testimony was consistent with a joint report issued by the FBI, the CIA and the National Security Agency in January, which concluded: “We assess Russian President Vladimir Putin ordered an influence campaign in 2016 aimed at the U.S. presidential election. Russia’s goals were to undermine public faith in the U.S. democratic process, denigrate Secretary Clinton, and harm her electability and potential presidency. We further assess Putin and the Russian government developed a clear preference for President-elect Trump.”
A key question is whether Russia merely preferred Trump or worked with members of his campaign. Comey confirmed in previous testimony March 20 that the FBI was investigating the possibility of collusion between the campaign and Russia.
Former CIA director John Brennan testified May 23 that his agency alerted the FBI last year to a troubling pattern of contacts between Russian officials and Trump associates.
“I was worried by a number of the contacts that the Russians had with U.S. persons,” Brennan said.
Trump associates reportedly under scrutiny include Paul Manafort, who was Trump's campaign manager, and former advisers Roger Stone and Carter Page.
Possible attempts to obstruct justice
Comey testified last week that while he was still head of the FBI, he told Trump on three occasions that the agency was not investigating him, individually. “Officials say that changed shortly after Comey’s firing,” The Post reported Wednesday.
The timing of the FBI's decision to investigate the president for possible obstruction coincides with Trump's admission, in a May 11 interview with NBC News, that he was thinking of the Russia investigation when he terminated Comey.
“In fact, when I decided to just do it, I said to myself, I said, ‘You know, this Russia thing with Trump and Russia is a made-up story,' " Trump told Lester Holt.
A second potential concern for Mueller relates to actions Trump may have taken to impede a separate FBI investigation of former national security adviser Michael T. Flynn. In his most recent testimony, Comey described a one-on-one conversation in which the president allegedly said, “I hope you can see your way clear to letting this go, to letting Flynn go.”
Comey testified that he “understood the president to be requesting that we drop any investigation of Flynn in connection with false statements about his conversations with the Russian ambassador in December.”
The Post reported last month that Trump, in addition to his direct appeal to Comey, asked Daniel Coats, the director of national intelligence, and Adm. Michael S. Rogers, the director of the National Security Agency, to help persuade Comey to let Flynn go. Coats and Rogers have agreed to be interviewed by Mueller's team, according to Wednesday's Post report.
Possible financial crimes
We know less about this prong than the other two. The Post reported last month that “in addition to possible coordination between the Kremlin and the Trump campaign to influence the 2016 presidential election, investigators are also looking broadly into possible financial crimes — but the people familiar with the matter, who were not authorized to speak publicly, did not specify who or what was being examined.”
In the same article, The Post reported that investigators are scrutinizing meetings held by Jared Kushner, Trump's adviser and son-in-law, including a December sit-down with Sergey Gorkov, the head of Vnesheconombank, which the Obama administration sanctioned after Russia’s annexation of Crimea and its support of separatists in eastern Ukraine.
Talking to the head of a bank under U.S. sanctions is not a crime, on its face, and Kushner has not been accused of wrongdoing.
Investigators also are examining the dealings of Flynn. A grand jury in Alexandria, Va., has issued subpoenas for records related to Flynn’s businesses and finances. A company owned by a Turkish American businessman who is close to top Turkish officials paid the Flynn Intel Group more than $500,000 for research on Fethullah Gulen, the cleric that Turkish President Recep Tayyip Erdogan claims was responsible for a coup attempt last summer.
In March, Flynn retroactively registered with the Justice Department as a paid foreign agent for Turkish interests.