North Carolina v. Alford concerns a situation back in 1963 where a man named Henry Alford was accused of killing another person.  There were no eye witnesses to the actual crime, but before the other man was found murdered Alfrord had taken his gun, declared to his friends he was going to kill him, and come back later saying that he had killed him.  Based on this, it was pretty certain that Alford was going to be found guilty.

Alford claimed he was innocent, but if convicted he faced North Carolina's death penalty.  His lawyer convinced him to plead guilty to a lesser crime, second-degree murder, in order to avoid this.  Alford did plead guilty to second-degree murder, but made it known that he was still innocent and that he was doing it to avoid the possibility of being executed.  Later, he would say:

"I just pleaded guilty because they said if I didn't, they would gas me for it."

That is the crux of the case. It's similar to a "no contest" plea, in which a defendant accepts a punishment without admitting guilt, but in Alford's case he was essentially pleading guilty while asserting his innocence at the same time.  Based on his guilty plea the judge in the case sentenced him to 30 years.

It would have been over at that point, but Alford appealed.  He claimed that his rights had been violated because the threat of the death penalty had, in essence, coerced a guilty plea out of him.  Based on North Carolina law (at the time at least, I don't know if they still do this,) a person could receive the death penalty only if they pleaded innocent and were found guilty.  Pleading guilty could still get a person a life sentence, but not death.

The case worked it's way up to the federal Supreme Court as lesser federal courts found in favor of Alford, (though state Supreme Court did not.) The federal courts were concerned that Alford's plea had been motivated by fear and was therefore not valid.  The Supreme Court ruled in favor of accepting Alford's original plea, siding with the North Carolina Supreme Court.  They did it for a few reasons:

• Alford had a competent lawyer,
• Alford's interests did lay in the direction of a guilty plea,
• There was strong evidence of guilt.

The ruling essentially allows for a defendant to enter a guilty plea while still saying they are innocent.  That's the importance of the case. As long as the requirements for a good lawyer and good evidence are met, a judge can accept such a plea.

[I have attached below an interesting bit from the "Principals of Federal Prosecution."  You'd be interested in the bit: "9-27.440: Plea Agreements When Defendant Denies Guilt."]