While the Supreme Court upheld the arrangement as constitutional, critics said it permitted a prosecutor to run amok. Republicans learned to hate the arrangement during the Iran-contra investigation into the Reagan administration, and Democrats did, as well, during the Whitewater and Monica Lewinsky investigations into President Bill Clinton. When the law expired in 1999, Congress did not renew it.

What would the appointment of a ‘special counsel’ do?

This position dates to 1999, when the Justice Department issued new regulations to create it after the independent counsel law expired. Special counsels are empowered to run an investigation with greater autonomy than a United States attorney normally enjoys. The regulations say special counsels “shall not be subject to the day-to-day supervision of any official of the department.” A special counsel also generally decides on his or her own “whether and to what extent to inform or consult with the attorney general or others within the department about the conduct of his or her duties and responsibilities.”

But if Mr. Rosenstein were to appoint one, the special counsel would still be ultimately subject to his control — and Mr. Trump’s. That means the special counsel’s decisions could be overruled, and he or she could be fired.

What does it mean that Mr. Rosenstein is in charge?

Generally, United States attorneys directly oversee criminal investigations, working with the head of the National Security Division in counterintelligence matters. But in a case that raises politically delicate issues, the attorney general would normally be briefed on proposed major decisions — like whether to subpoena a high-profile witness or bring charges — and could overrule them. With Attorney General Jeff Sessions recused, because of his meeting as a senator with Russian officials, Mr. Rosenstein is currently exercising that role.

What about a special congressional inquiry?

At least two Republicans on Tuesday cited the Comey firing in calls for Congress to create a special body to investigate the Russia matter, instead of or in addition to the investigations by the House and Senate Intelligence Committees.

Representative Justin Amash, Republican of Michigan, said he was studying legislation that would create a special commission. And Senator John McCain, Republican of Arizona, called for the creation of a special select committee, which would be made up of members of Congress.

“I have long called for a special congressional committee to investigate Russia’s interference in the 2016 election,” Mr. McCain said. “The president’s decision to remove the F.B.I. director only confirms the need and the urgency of such a committee.”

Such panels would have subpoena power, but their work would culminate in a report. They would not have the power to bring criminal charges.