Yesterday, the Obama administration invoked executive privilege to prevent the release of certain documents to Congress related to Operation Fast and Furious, the arms-trafficking sting gone awry that came to light last year. (As we've detailed, federal agents lost track of hundreds of guns sold to suspected gun smugglers, many of which later turned up at crime scenes in Mexico).

The fall-out from the failed operation has been an ongoing battle between Attorney General Eric Holder and congressional Republicans, in particular Rep. Darrell Issa, R-Calif., chair of the House Committee on Oversight and Government Reform. Issa wants documents related to the Department of Justice's investigation of the operation.

The committee voted yesterday to recommend that Holder be held in contempt of Congress for not turning over some documents. Holder says that his office has already released thousands of documents, and that the others that Issa wants are internal communications protected by executive privilege.

In the midst of all this back-and-forth, we lay out exactly what the executive privilege is, and what it means in this case.

So what is executive privilege?

The president can invoke executive privilege in order to withhold some internal executive branch communications from the other branches of government. The privilege is based on the separation of powers between the branches.

Executive privilege has been invoked since the U.S.'s early days but isn't in the Constitution. It was only in 1974, when Richard Nixon tried to prevent the release of White House tapes during the Watergate investigation, that the Supreme Court upheld its constitutionality, and set some parameters for it. The Court ruled that no claim on executive privilege is absolute, and can also be overcome if evidence is needed in a criminal trial. (For a full legal history, see this report from the Congressional Research Service.)

So what does it usually cover?

Various administrations have set their own policies as to when they can invoke the privilege. (The Washington Post has a handy timeline showing when presidents have used it.)

Bill Clinton used them a lot, 14 times during his presidency. In 1998, his attempt to keep White House aides from testifying about the Monica Lewinsky scandal was struck down, the first time since Nixon that executive privilege was overruled in court. George W. Bush invoked the privilege six times, not always successfully.

Legal challenges have established two general categories of executive privilege: presidential communications and deliberative process.

The presidential communications privilege applies to communications involving the president or his staff that immediately pertain to the president's decision-making process. The idea, according to Mark Rozell, a professor at George Mason University, and author of a book on executive privilege, is that "the president should have the right to candid advice without fear of public disclosure."

Deliberative process involves a broader scope of executive branch activity: discussions involving White House staff or within other agencies on legal or policy decisions that don't necessarily involve the president or his immediate advisers. Again, the argument is that government officials need to feel like they can talk honestly. The deliberative process privilege, Rozell says, is generally easier to challenge than a claim of presidential communications privilege.

What is Obama's view on executive privilege?

In 2007, in an interview with CNN, then-Sen. Obama criticized "a tendency on the part of [the Bush] administration to try to hide behind executive privilege every time there's something a little shaky that's taking place." During his presidential campaign, Obama said that executive privilege "generally depends on the involvement of the president and the White House," referring, presumably, to the narrower, presidential communications privilege.

Yesterday, the White House emphasized that this was the first time Obama has invoked the privilege.

So what is Holder trying to keep secret?

Holder asked Obama to invoke executive privilege over documents having to do with "the Department's deliberative process concerning how to respond to congressional and related media inquiries into that operation." In other words, Holder is saying the documents don't deal with the government's actual response to Fast and Furious.

The letter doesn't explicitly mention presidential communications. So, contrary to a few Republican congressmen's claims, the communications could be just between Justice Department officials and not include anybody at the White House itself.

What happens next?

Nobody's quite sure.

Holder could be voted in contempt of Congress by the Republican-controlled House. But if he is, it's not clear Holder would be forced to do anything. Congress could file a lawsuit against him or try to arrest or fine him to force him to comply. But that's a long shot. Disputes over executive privilege are usually settled with a compromise between the executive branch and legislatures.

In 2008, two of President Bush's advisers were held in contempt of Congress for refusing to comply with an inquiry into the firing of several U.S. attorneys by citing executive privilege. This led to a stand-off, with the attorney general telling the local U.S. attorney not to enforce the contempt citation. Eventually, more than a year later, a deal was reached with Congress and the issue was dropped.

If it did go to court, the deliberative process privilege might not hold up. In 2004, Bush asserted presidential communications privilege over Clinton administration documents relating to pardons. A court ruled that they didn't immediately involve the president and would be protected, "if at all" by the deliberative process privilege. The court then ruled that they weren't protected under that privilege — and the Pardon Attorney had to turn over more than 4,000 documents.