OP here with bit of an update:
I called HR this morning, but the director had a vacation day. She will be in tomorrow. I spoke with the HR assistant for some time, and explained the situation. She was very helpful and pulled the file for the position. My resume was not in there. She asked if I got confirmation about applying. I told her I did - from the director. She was very , very puzzled and said she really couldn't help me anymore, and that I would need to speak with the director personally since she (the assitant) did not understand why my resume wouldhave not been included. The assistant left a message for the director to call me tomorrow when she gets in.
To me, it really just sounds like the director of HR messed up. She forgot to put my resume in the file, and I was left out.

Looking forward to hearing the HR director's take on this.
Additional info - as someone else pointed out, I have young children. I left my f/t job upon the birth of my first child, and accepted this position 6.5 years ago. I previously worked 25 hours a week (nights/weekends), but when I was expecting my 3rd child, my manager and I agreed to a new schedule. So since the birth of my youngest (now 5), I have worked just 5.5 hours a week (Saturdays).
Now that the kids are getting older, I am looking for a new situation. One of the main reasons I kept my little 5-hour job is to be able to move into something with more hours when the time was right. I have been watching the openings for nearly 2 years waiting for the right position, and then I applied.
My DH and I talked about this at length this weekend. We do not feel that this is an organization that I really want to work for anymore. DH is interested in looking into some side jobs in his field that would replace my little salary, and I am going to focus on either finding a full-time position or continuing with my schooling (I have been taking college classes for a year, with the intention of getting my teaching cert - but education is not a great field to try to get into right now).
I am going to talk with HR, get some answers, and hopefully see that someone higher up checks on the HR director to make sure she is doing her job as she should be. I do not have summer hours, so I am on break from my job right now. I will most likely be giving my notice this summer and making a break from this company.
I am going to request a time to view my personnel file as well.
At this point, I just want to get to my vacation in 20 days!