But the ball of water nearest the moon experiences a somewhat stronger moon-gravity than the other objects and it falls faster toward the moon. Similarly, the ball of water farthest from the moon experiences a somewhat weaker moon-gravity than the other objects and it falls more slowly toward the moon. As a result, the earth and the other ball of water pull away from this outer ball so that this ball bulges outward, away from the earth. It's interesting to note that the earth itself bulges slightly in response to these tidal forces. However, because the earth is more rigid than the water, its bulges are rather small compared to those of the water.
Lunar gravity is partly what causes oceanic currents.
As the earth turns, these bulges move across its surface and this movement is responsible for the tides. If there were more than one moon, the tidal bulges would become misshapen. That is essentially what happens because of the sun. As the moon and sun adopt different arrangements around the earth, the strengths of the tides vary. Extra moons would probably just complicate this situation so that the strengths of the tides would vary erratically as the moons shifted their positions around the earth.