Orbits
The Sun is a star and the center of our solar system. Gravity causes the solar system's celestial bodies --- including the Earth --- to revolve around the Sun. Most celestial bodies' orbits are oblong (elliptical) as opposed to being a perfect circle (spherical). Because of their elliptical orbit and position in space, as the Earth revolves around the Sun, the moon revolves around Earth. Since the moon revolves around the Earth, the moon, by default, also travels around the Sun while orbiting Earth.
Eclipses
During an eclipse, the appearance of one celestial body is obscured by another passing celestial body and/or its shadow. A solar eclipse occurs when the moon becomes aligned between the Earth and Sun. A lunar eclipse takes place when the moon travels behind the Earth and becomes aligned in the middle of the Earth's shadow. The Earth's shadow is on the side of the Earth that is facing away from the Sun at a given time.
Comparison
In a solar eclipse, the object that is obstructing the Sun's rays of light, creating a shadow, is the moon. In a lunar eclipse, the object that is obstructing the Sun's rays of light, creating a shadow, is the Earth. Since the Earth is much larger than the moon, it creates a much larger shadow --- and umbra --- during a lunar eclipse, than the umbra created by the moon during a solar eclipse.
Moon Phases
The moon is the Earth's only natural satellite and is called a dwarf planet, by some experts. It's approximately 2,160 miles in diameter and its journey around the Earth takes 29 1/2 days. As the moon orbits around the Earth, it changes positions on its axis, giving people on Earth a different view of its surface at various times. These different surface views are called moon phases, which include the new moon, new crescent, first quarter, full moon, last quarter and old crescent. The full moon phase is when the entire front surface of the moon is fully visible from Earth. Lunar eclipses only occur during the full moon phase.