There is only one full moon each month because the moon only opposes the sun once per month in its orbit. The phases of the moon are caused by how much of its visible surface is illuminated by the sun's light, and a full moon only occurs when the moon is on the opposite side of the Earth from the sun.
As the moon travels around the Earth, its phase changes. When the moon is on the same side of the planet as the sun, its illuminated surface is facing away from Earth, causing the visible portion of the moon to appear completely dark. As it orbits, more of its illuminated side becomes visible, transitioning from the dark "new moon" to a "crescent," "half" and "gibbous" moon. Only when the moon is in direct opposition to the sun is the entire surface illuminated, producing a full moon. Since this only happens once per orbit, there is only one full moon per month.
Full moons are also the only time lunar eclipses occur. The moon must be in a particular arrangement with the Earth for the Earth's umbral shadow to darken the moon's surface, and even then refracted light through the atmosphere gives it a dark reddish glow instead of obscuring it completely.