The first thing I'd like to clarify is this (both entries are from Wikipedia):
Astronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere. It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the formation and development of the universe.
While Astrology is a group of systems, traditions, and beliefs in which knowledge of the apparent relative positions of celestial bodies and related details is held to be useful in understanding, interpreting, and organizing information about personality, human affairs, and other terrestrial matters.
Simply put, Astronomy is the application of science (physics, chemistry, biology) to objects out in space, while Astrology uses the positions of the stars/planets/sun 2000 years ago to tell you something that could apply to almost anyone. It is true that, back in the good old days, these two were indistinguishable. That is no longer the case - so please keep in mind if you ask an Astronomer if they study Astrology, make sure you know how to bob & weave your way out of a slap to the head.
In Astronomy, there are two broad research areas: theoretical and observational. Many people choose one or the other, but some do overlap. We couldn't do one without the other. Most spend a lot of time in front of a computer: theorists write code, observers reduce and analyze data.
Inside each broad area are the same sub-fields which are basically classified by object (starting small and working our way up): meteors, asteroids/comets, planets, stars, star systems, star clusters, galaxies, galaxy clusters and the whole frickin' universe. The further out you go, the more theoretical things get.
A common misconception in Astronomy is that we've been studying it for thousands of years, so we must already know everything. This is far from the truth, even with objects close by like the Moon or Mars. The reason why we don't know much is because we can't just go over to the comet/star/galaxy we're studying, take a sample and analyze it back in the lab. We have to come up with innovative ways to learn about these objects that are so far away.
Even though that seems to be the downfall of Astronomy, it's really the reason why our research is so important. Without Galileo building his telescope to look at Jupiter and Saturn, we wouldn't have TVs, cameras, photocopying machines or anything else that depends on optical equipment. Without understanding the motions of the planets, we wouldn't have satellites or GPS. Plus, without the astronaut program, we wouldn't have Tang!
I find I get caught in the problem many Astronomers know: what are we doing that will really help society? I try to keep in mind that it's the pure scientific research that leads the way for engineers or developers that take those ideas and make them into useful, everyday things.
Personally, Astronomy helps me reach students of all ages because of the common awe and wonder it inspires - everyone has looked up at the stars and wondered about what (or who) is out there. It's a romantic, sexy science, and if people are already interested in it, then it's as easy as 1-2-3 to take them to the next level.
Hmm - I seem to have veered away from the specifics on Astronomy research, but I hope that I shed some light on to why Astronomy is important and why I love it so much.