I've seen many different types of anime girls but the two prominent ones that have stuck to my memory are yandere and tsundere girls. Now everytime I think about a normal relationship, the image of a boring and cliche girl comes up. I just can't stand normal girls in an anime! Haruna from To Love-Ru for example.
Yandere is a behaviour (not limited to girls) in a romantic relationship where the guy or girl loves the partner so much that obsessed would a more suitable term.
Tsundere is when the guy/girl's behaviour in a relationship consistently alternates between cold and shy/warm. You can see an example of this in Zero No Tsukaima.
Sometimes I wonder if anime and manga has affected my real life preferences for girls.
Here's a picture I dug up from google images to help your understanding.