A briefcase that travels with the President of the United States when he is away from the office. The President doesn't carry the briefcase, but instead a military aide is always near the President with the magic briefcase handcuffed to him. If I remember correctly, it is always a marine.
The briefcase contains a secure satellite communications system and authentication codes. The codes ensure that the person on the other end of the line knows that they are talking to the President. Once everybody is sure who is talking to whom and makes a little small talk, the President tells them to blow shit up in his own special secret way.
The aide has on a few occasions been separated from the President for a few moments. This most famously happened when President Clinton left a NATO summit without his aide.
The name Nuclear Football comes from the original code-name of the project, which was "Drop Kick." There have been several versions, although exactly how many is unknown.
On a side note, this is what I thought the node Man's desire to blow shit up, and to have a nice attache case was about.