The point of being a professional astronomer, in my experience, is to contribute understanding of what the universe is, how it is structured, how it came to be, and what its future might hold.  Most likely this is because the first astronomer I ever met was a professor of cosmology.  I've come to accept that he's probably the reason my main research interest tends to observational cosmology.  Of course, this is a very broad and relatively unhelpful answer to the astronomer question.  Sure, that's the intention, but how do we get there?

I think it's safe to assume that the journey to becoming a professional astronomer begins as an undergrad, or if you're very lucky, as a high school student.  I think mine was a bit of both, as I did get the opportunity as a junior to do some busy work for he of blog title fame.  But that was a week long, and although it was some exposure I doubt it's how careers in astronomy start.  Careers in astronomy, at least for a Caltech student, probably start with a SURF fellowship.  I know SURF was my first real look at what an astronomer does.  I sat alone in an office 8-9 hours a day, writing code in a language I'd never seen before to analyse data I didn't understand, and I had fun doing it.  I think that enjoyment that's what sets the astronomer apart from the average person.

Then the natural course of things is to go to graduate school.  This is where you decide What You Want To Do With Your Life.  As far as I know, you don't have to decide right away.  Unless you're in the UK in which case you need to know what you want to study before you've learnt anything about it.  At least, that is what my SURF mentor who is English tells me.  Being a grad student requires doing semi-independent research under the guidance of a faculty member who works on a similar topic.  You'll probably start to hate your field at some point during this process, but hopefully you'll get over it soon.  Next is the postdoctoral fellow.  I have no idea what a postdoc does.  Don't tell my SURF mentor that because he is one.

I believe your career options then become a) professor at an academic institution, b) research scientist some place like the SAO, or c) finance.  I'm sure there are more options, I'm just uneducated in that side of things.  The first two options strike me as pretty similar except the professor track astronomer will probably have to teach at some point or another.  This part is where you get to move on to independent research in topics that interest you.  You might find out something that only you know about, and that's a rewarding experience.  Although any work in astronomy is rewarding if it's what's truly exciting and inspirational to you.

I have given my impression of what it takes to become an astronomer.  So once again, we're back to the question of what does it mean to me to be an astronomer?  It's going to take a lot of work.  Astronomy, as it turns out, is hard.  But the work will be worthwhile because I'll be learning how the universe works, or how we think the universe works.  Maybe I'll end up amending some of that knowledge.  Who knows?  Being an astronomer means getting excited about the mysteries of space and our tiny place in it.  It means realising how small we really are in the grand scheme of things, accepting that, and moving on to understand why.  Most of all, it means that when your friend starts talking about M83 and means the band, this is all you can see.