Every cover I make bows within a week, I keep hearing about grain direction and I do not understand what it is doing

The mechanism is simple and once you have it, every warping problem you will ever have becomes predictable.

Paper is made of fibres that mostly lie in one direction, the direction the sheet travelled during manufacture. That is the grain.

Fibres swell across their width far more than along their length. So when paper takes on moisture, from glue, from paste, from the air: it expands noticeably across the grain and barely at all along it. It is not a small difference; it is several times.

Now your covers. A cover is board plus paper plus whatever is inside, glued together. When you apply wet adhesive, one layer wants to expand across its grain. If the layers disagree about which direction that is, they fight, and the assembly curls toward whichever side is pulling harder. Then as it dries and shrinks, it curls again, sometimes the other way - which is exactly your inconsistent result.

The rule that falls out of it: grain runs parallel to the spine, in every component - the board, the covering paper, the endpapers, and the pages. Then everything expands and contracts in the same direction, and instead of fighting, they move together. The spine can accommodate that movement; the width cannot.

That is also why a book with the grain wrong feels stiff and refuses to open flat. The pages are resisting the fold in the direction they least want to bend.

How to actually find the grain, since that is the practical half and nobody shows you.

Bend test. Hold the sheet and bend it gently into an arc one way without creasing, then the other. It bends more easily and with less resistance along the grain. This is the quickest and works on most weights.

Tear test. Tear a small piece off a corner in each direction. It tears straighter and more cleanly along the grain, and raggedly across it. Destructive but unambiguous, and the one I trust on unfamiliar stock.

Damp test. Wet a small strip on one side and it curls; the axis it curls around tells you the grain. Best for board, where bending is hard to judge.

On board specifically: greyboard and millboard both have grain and it matters just as much as in the paper. Suppliers often state it, and the sheet size convention usually tells you: the second number of the stated dimensions is conventionally the grain direction, but confirm rather than assume, because it is not universal.

When buying, the phrase you want is short grain or long grain relative to the sheet, and it is worth asking. Buying the wrong one means cutting your sheets the wasteful way to get the grain right, which is annoying but far better than the alternative.

Two things that reduce warping even when the grain is correct, because grain is necessary and not sufficient.

Counter-line everything. Whatever you glue to the outside of a board, put something comparable on the inside. A cover paper on one face and nothing on the other means one side is pulling and the other is not, and it will bow no matter how the grain runs. That is what the paste-down is doing structurally, beyond looking tidy.

Press until properly dry, not until it looks dry. Under weight, between boards, overnight at minimum, and longer for anything heavily pasted. Most of the movement happens as moisture leaves, so a cover taken out of the press after an hour has done most of its warping in front of you. Changing the blotting paper partway through helps a lot.

And the reason yours look fine and then curl days later: they were still losing moisture when you took them out. The press was holding them flat while they dried unevenly, and once released they moved.

One more: use less adhesive. A thin even coat introduces far less water than a generous one, and most beginner warping is a moisture quantity problem sitting on top of the grain problem. Thin coat, correct grain, counter-lined, pressed overnight - do all four and it stops.