Twilight
I got this series for my birthday (THANKS, Jav!) and I loved, loved, loved it! In fact it pretty much consumed my life for the days (ok, weeks!) following my birthday as I devoured each one.
The first book in the series is Twilight and it introduces you to the saga that is Bella and Edward's relationship. This is a very girly vampire story. In fact most of the story is based on Bella's analyzing and reanalyzing...agonizing and re-agonizing...worrying and re-worrying. :)
It is very suspenseful but in no way would I classify it as "horror" so don't be afraid to give it a try.