What is regular heart rate? BPM?

It's generally around 60bpm for women. The more fit you are, the slower your heart rate will be. That's because the heart doesn't have to work as hard at rest. It sounds like you're probably in good shape. You might not be pushing yourself hard enough during your runs, unless by run you mean a brisk walk or jog.

Look at Nike's heart rate monitors so that you can accurately gauge your heart rate.

"normal" is between 60-100. If you are physically fit, your normal resting heart rate may be lower. When you are physically active it can be much higher. Heart rate is also relative to your blood pressure. When your blood pressure is up, your pulse will be down, and vice versa.

That is fine, but I would expect someone running to have a much higher rate if they were really pushing themselves - are you sure you are measuring it correctly?