During a solar eclipse, the sun is eclipsed, or blocked from view. This occurs because the moon passes between the sun and the earth. Solar eclipses can be partial, total, or annular, a form of eclipse where the central disk of the sun is blocked but the outer edge, or corona, stays in view.
During a lunar eclipse, the moon is eclipsed, or blocked from view. This happens when the earth passes between the sun and the moon, and casts a shadow on the moon. The only time this can occur is during the full moon phase, when the sun, earth, and moon are in a straight line relative to one another.
Neither type of eclipse happens terribly often because the orbits of the earth and moon are elliptical rather than circular, and the orbit of the moon sits at an angle relative to the orbit of the earth around the sun. Since an eclipse only occurs when the alignment is perfect, we only have them once in a while.