That's a great question! Tides are caused by tidal forces, and the answer to your question lies in the definition of a tidal force. A tidal force is related to gravity, but it isn't the same thing. It's really the difference between the the strength of gravity at two locations.
The gravitational attraction between two objects (say the Earth and the Moon) decreases with distance. This means that the Moon's gravity pulls most strongly on the side of the Earth closest to the Moon and least strongly on the side of the Earth farthest from the Moon. Tidal forces on the side of Earth closest to the Moon pull material (mostly water) toward the Moon. Tidal forces on the other side of Earth actually pull material away from the Moon. The resulting deformation of Earth looks the same when the moon is at opposite sides of its orbit, like full moon and new moon or first quarter and third quarter, as shown in the diagram on this page. That's why tides around the equator are higher during both a new moon and a full moon (spring tide).
The Sun also affects the Earth's tides. However, tidal forces due to the Sun are about half as strong as those due to the Moon. This seems strange, because the Sun's gravity at Earth is much stronger than the Moon's. But remember that tides concern the difference between gravity's pull at opposite sides of the Earth. The radius of the Earth is a very small fraction of the distance between the Sun and the Earth, about 0.005%. As a result, the difference between the Sun's gravitational pull on either end of the Earth is small. In contrast, the radius of the Earth is about 1.7% of the distance between the Earth and the Moon. So even though the Moon's gravity isn't as strong as the Sun's, lunar tidal forces are stronger than solar tidal forces, so lunar tides are stronger than solar tides.
