The Moon makes one trip around the earth (west-to-east) in a little over 27 days. But, once again, there is a difference between the one trip around with respect to the stars (the sidereal month), and one trip around with respect to the Sun (the synodic month). As we saw when considering the Sun, the synodic month (i.e., New Moon to New Moon) is longer, about 29 days.
The Moon does not orbit the earth in quite the same plane as the Sun -- the circle it takes on the sky it tiled from the ecliptic by about 5 degrees. Hence the Moon spends 1/2 its time just slightly north of the ecliptic, and 1/2 its time slightly south of the ecliptic (but always in a zodiac constellation). Twice a month, the Moon's path crosses the ecliptic. If the Moon happens to cross the ecliptic at the exact spot the Sun is, the Moon will block out the Sun's light, and we'll have a solar eclipse. (This does not happen often. The Moon is rather small and casts a small shadow. Consequently, it really does have to make a bull's eye with the Sun. Most of the time, its shadow misses the earth, and even when the shadow does reach the earth, it covers only a small area.) If the Moon blocks out the Sun completely, we call it a total solar eclipse. If it only blocks out a section of the Sun, then it's a partial solar eclipse. Finally, because the Moon's orbit around the earth is not perfectly circular, sometimes it will be perfectly aligned with the Sun, but still the solar disk will poke out around the edges. Then we have an annular solar eclipse. Because the Sun is so bright, unless it is totally eclipsed, it will be too bright to see with your eye. In fact, in most cases, you'll barely notice anything happening at all! Total eclipses are very rare, totally spectacular, and only last a couple of minutes. If you can, take a trip to see one. (But be prepared for crowds along the path of totality.)
About twice a year, the Moon will cross the ecliptic at a location exactly opposite that of the Sun. When this happens, the Moon will get in the Earth's shadow. This is a lunar eclipse. (The earth is substantially bigger than the Moon, so it's shadow is bigger. Hence lunar eclipses aren't all that rare.) Although the Moon is being eclipsed, it does not disappear completely. Some sunlight (mostly red light) makes it through the earth's atmosphere to the Moon. Consequently, when the Moon is eclipsed, we see it as a dull, red orb.