Do your research.
Observe the situation carefully before attempting to talk to the girl to find out a little about her. If you see her in the grocery store, look at what's in her cart to find out the kinds of food she eats. If you see her in a bar, look at what drink she ordered. This will help you know a little about her going in, which can help spark up a conversation.
Talking to strangers can be unnerving, but try to hide your nerves as best you can. If you walk up to her and appear scared and uncomfortable, chances are she will feel uncomfortable as well. Tell yourself that this girl is lucky to meet you.
Approach her by pointing out or asking about something you observed.
If at the grocery store you notice that she has a lot of vegetables in her cart, ask her if she's a vegetarian, and mention that you have thought about becoming one yourself. If she orders a cocktail that you don't recognize, ask her what kind of drink she ordered. The main thing to remember is to have something to say. There is nothing worse than approaching a girl and then staring at her awkwardly as you try and think of something to talk about.
Compliment her, but only if it's genuine.
If you think she has pretty eyes or a nice smile, it is alright to tell her so. Just make sure that the compliment is sincere, because women are very aware if a compliment is false. You don't want to come across as phony.
Talk about things that are relevant to you and that you are passionate about.
Your excitement on the subject will come across as friendliness. Just make sure not to dominate the conversation.
If she brings up a topic that you don't understand or are unfamiliar with, it's OK to be honest. Better yet, ask her about it. If she mentions that she loves 17th century art, and you don't know anything about it, ask her to explain it to you. She'll be delighted to talk about something that is of interest to her.
Don't feel bad if the conversation does not seem to be going anywhere or the connection is not there.
Even if you don't meet the girl of your dreams the first time around, it is always helpful to gain experience at talking to girls. You will feel much more comfortable the second or third time around.