the tidal force
Gravity does not just pull an object as a whole; because it weakens with distance, it pulls the near side of a large body a little harder than the far side. That difference in pull across the body is the tidal force, and it tends to stretch the body along the line to whatever is attracting it. It is why the oceans bulge and why the sea rises and falls twice a day.
The mechanism is the whole point of the name difference. The Moon pulls the water on the near side of Earth strongest, the solid Earth as a whole a bit less, and the far-side water least of all. Relative to the Earth's centre, the near water is pulled toward the Moon and the far water is, in effect, left behind, so two bulges form, one facing the Moon and one opposite. Because it is the difference of an inverse-square pull, the tidal force falls off much faster, roughly as 1 / r^3, so nearness matters enormously.
That r^3 dependence explains a surprise: although the Sun pulls the Earth far harder than the Moon does overall, the Moon raises bigger tides, because the Moon is so much closer that its pull changes more sharply across the Earth. The Sun still helps: when Sun and Moon line up we get large spring tides, and when they are at right angles we get gentler neap tides. Taken to an extreme, tidal forces near a black hole are so violent they would stretch a falling object into a thin strand, an effect nicknamed spaghettification.
The Moon's pull on Earth's near-side water is only a fraction of a percent stronger than on the far side, but that tiny difference, acting over a whole ocean, lifts tides of a metre or more, twice each day as the Earth turns beneath the two bulges.
Tides come from the tiny difference in gravity across the Earth, not from the total pull, and there are two bulges, hence two high tides a day.
Tidal force is the difference of gravity across a body and falls off as about 1/r^3, so the nearer Moon out-tides the Sun despite the Sun's far stronger overall pull.