How can a piece of A4 paper be folded in exactly three equal parts?

Solution 1:

Fold twice to obtain quarter markings at the paper bottom. Fold along the line through the top corner and the third of these marks. The vertical lines through the first two marks intersect this inclined line at thirds, which allows the final foldings.

(Photo by Ross Millikan below - if the image helped you, you can up-vote his too...)

Solution 2:

Here is a picture to go with Hagen von Eitzen's answer. The horizontal lines are the result of the first two folds. The diagonal line is the third fold. The heavy lines are the points at thirds for folding into the envelope.