The basic physics
The Moon pulls the water on the side of Earth facing it. For the same reason, the water on the opposite side is also "left behind" while the rest of the planet accelerates closer to the Moon, forming a second bulge there. Earth rotates under those two bulges, and that explains the two daily tides.
The Sun also pulls, but with less force (about 46% of the Moon's pull, even though the Sun is much larger, because it's much farther away).
When the Sun, Earth, and Moon are aligned (new moon or full moon), the forces add up. The tides become wider. That's the spring tide regime (also called syzygy tide).
When the Sun and Moon are at a 90° angle as seen from Earth (first and last quarter), the forces partly cancel out. The tides become milder. That's the neap tide regime (also called quadrature tide).