Social settings are usually a good way to get to know people, but when someone is less than socially gregarious, it becomes a bigger challenge. Introverted people have a lower than average tolerance for interaction with strangers and may find social gatherings exhausting. In order to better get to know an introverted person, display some understanding for the challenges introverts face while becoming acquainted with someone.
A crucial thing to remember about getting to know an introvert is that it's important to give her time. While you're still a stranger, it's easy to exhaust an introvert by pushing too much interaction on her. Understand that it's not personal or a reflection on you -- it's just that introverts can feel uncomfortable with people they don't know well and that discomfort is draining for them. Don't avoid talking to an introvert, just limit your interactions a little more and give the person some time to process things between interactions. As time goes on and the introvert gets to know you, this won't be such and issue.
In social settings, give an introvert a chance to get to know you without being required to interact too much. Most introverts tend to be quiet and reserved around strangers, but that doesn't mean they aren't engaged in the social activities as observers. Help an introvert get to know you by giving him chances to be nearby when you're having meaningful conversations with others at social gatherings. Make sure, however, that what you're doing isn't obvious; have the conversations for their own sake, but look for chances to do so when the introvert is nearby.
When you feel that you've gotten to know an introvert well enough to initiate a more familiar level of interaction, start with written communication. Send the introvert letters, text messages or emails that she can respond to at her leisure. Make it clear that you don't expect an immediate response and avoid inundating the person with too much information or too many questions. This allows her to give you her time, attention and mental energy on her terms, when she feels able to do so.
Once you've broken the ice with the introvert and start to see him warm up to you a little more, engage him in some meaningful conversation. Avoid superficial or “safe” topics and move directly into areas that are actually related to things you have in common and find meaningful. Introverts tend to find “small talk” exhausting and difficult and prefer to avoid it.