What do graduate programs look for in an applicant? How does admissions work? 

Different programs differ immensely. I imagine that what got me into my graduate program only got me in because I happened to be taking a course from the department chair the quarter I applied to graduate school. My own grad school application would not get me into UCLA Biostat PhD program at this time, though it might get me into our MS program. [There's a pretty harsh moral in there somewhere.] Biostat programs often have less math requirements than do statistics program, but they also may or may not teach as much math stat as a stat program. Ours does teach plenty of math stat. A lot of programs (stat and biostat alike) admit to the MS, then pass you through to the PhD program if you do well. We admit a few people to the PhD, many more to the MS, and we admit to the PhD from our own MS. I believe this is how many programs work. (Another model is that some programs admit only PhD students, but if people don't make it through the PhD, they are sent off with an MS degree as a consolation prize.) There are no doubt other models.    

Is it more important that I take particular mathematics courses before sending in my application or get good grades in the ones that I do take?

I'd vote for good grades. A good grade when you've only taken one or two (upper division) math courses is an 'A'. If you're going for a PhD you'll need to show that you can do PhD work though, and that means taking a few difficult math courses. Undergrads usually don't take what I would call "really difficult statistics courses".  But that is certainly university/program specific. If you take a lot of math classes, then you have enough to average out the occasional bad grade. But don't ask me to tell you how to balance out the occasional bad grade (i.e. a B) more mathematics courses.  

What do you look for in a graduate student?

You need to put in many hours (Confer Outliers, by Malcolm Gladwell) to get really good at something. So putting in the time is worth while, starting now. For example, no graduate program teaches you everything about a subject; if you're going to have a well rounded education after receiving your PhD, you're going to have to teach yourself more than you learn in class. How do you teach yourself material? Start now, trial and error, learn stuff, do badly on somethings, better on others, and most importantly, keep on trying. So that's what I really want in a doctoral student: curiosity, an ability to figure stuff out, an inquiring mind, stick-to-it-tiveness, someone interesting we'd really like to have around for several years. Sadly that's not what is tested for on the GRE. 

Do you, as an admissions officer, look at the courses applicants are "currently enrolled in" even if there is not a grade attached to that course?

Yes, absolutely. But mainly when it matters to my admissions decision. Here's one hypothetical example: Consider someone with all B's in their first 2 years of college who suddenly 'finds themself' and gets all A's their junior year; if they are a math major, I really want to know about those Fall grades. If they are all A's that person could be considered for the PhD program. If they are a mix of B's and some A's, maybe I'm willing to admit to the MS program only. Someone with all A's in their first three years of undergrad (yes, they exist), I probably don't need to see senior year Fall grades to make an admissions decision.

Here's another common hypothetical: someone who qualifies for one degree program, but is enrolled in courses, that, if she gets A's in them, would qualify her for a different degree program. And suppose she prefers that 'different degree program'. Then I'm looking at the courses, and I actually need to wait to see the grades before I can make a sensible admissions decision. 

What math courses specifically do you look for in a student's application/ suggest I take before I apply?