The art of conversation is a complex skill that is often taken for granted, especially in the case of meeting a woman you're interested in. While there are many approaches to breaking the ice and starting a meaningful chat - confidence, respect and a relaxed, casual attitude are keys to a successful interaction.
Immediately after meeting a girl, start with basic introductory information. You may have been introduced to each other by someone else, but if you haven't, offer your hand to shake and say, "Hi, my name is **." If you've already been chatting for a little while, interject and say something like, "By the way, I'm** ." Next, say something by way of explanation about why you're in the current setting, if it's relevant, such as say, at a party, "I'm a friend of so-and-so's. We work together."
After introducing yourself, continue to break the ice by commenting on your current surroundings or goings-on. This might mean something as simple as talking about the weather, or commenting on any music you might be listening to or the food you're eating. If you're in a public setting, you might comment on the decor at a restaurant if it's noteworthy, or you might recommend a drink that the current bartender is particularly good at making. Whatever you choose, talking about your setting is a non-threatening way to start getting used to one another's company.
Once you've broken the ice, you can start making conversation that will help you start to get to know this girl. One of the best ways to do this is to ask her questions about herself. Ask questions that are meaningful but do not veer into personal or private territory. For example, it's appropriate to ask about things like work and hobbies, but not about subjects like ex-boyfriends (unless she brings it up and is casual about the subject). Start with questions like where she's from and how long she's lived in the area and what she does for a living. You can move from here into questions about what she enjoys doing in her spare time and what some of her interests and aspirations might be. Try to let the conversation flow organically and only introduce new questions if the chat starts to lull. Look for things you have in common and expand on these topics, but remember always to try to invite her to talk about herself whenever possible.
While it's a good idea to let her talk about herself a lot (most people enjoy talking about themselves and appreciate that someone else is interested in them), you'll need to talk about yourself as well. Volunteer information about yourself to echo the details she's sharing about her life, but only if she shows an interest. For example, if you ask her what her job is and she tells you, let her talk about herself as much as she wants and only mention what you do if she asks, or if it comes up as a part of what you're saying in response to her comments. Try to let her take up slightly more than half of the conversation and focus on areas of your life, opinions and interests that you have in common.
Talking About Yourself
Once you've laid down a strong foundation of common interests, you can move into expressing interest in pursuing her company further. You can find an excuse to get together at a later date, such as a business-related meeting, or you can ask her directly to accompany you on a casual get-together. If you don't feel ready for this, yet, you might just say that you hope you bump into her again, especially if you think it's likely that you will. This will lay the groundwork for more meaningful conversation the next time you see her.