The modern era has brought internet relationships to the forefront of the dating world. If you haven't given them a whirl yet, then you probably should. Meeting girls using the internet has many advantages over just waiting for a girl to walk into your life or having to say “Can I buy you a drink?” every other weekend. If you're worried that you might have to spend money or hop immediately into a commitment with someone you barely know, think again. Facebook is the premier 'dating site' that's not really a dating site, make use of it.
One of the best ways to meet girls using Facebook is to go back in time and connect with people from your past. No matter how many years out of high school you are, chances are that a large percentage of people from your senior class remember you, whether you were really close friends or not. High school is sort of a piece of youth that everyone tends to cling on to to some degree as they get older. Trying to talk to girls who you think might not remember you is worth a shot too. The beautiful thing about sending friend requests to people from your high school is that they'll probably take a glance at your profile, see that you have a couple common friends and that you went to the same school, and accept your request without thinking much else of it. It's a backdoor way into meeting some “new” girls without seeming like the sleazy internet stranger. The same can be done by seeking out girls you went to college with, but unless you genuinely knew them, your success rate will probably be lower as college membership is generally much larger than that of any given high school. If you know of someone specific from your past that you'd like to reconnect with it's as simple as typing a name into the search bar. The number of young adults using Facebook these days is constantly rising and maintaining an account as part of one's daily life is quickly becoming the norm.
You can also meet new girls on facebook courtesy of common friends. If a girl sees that you're friends with several of her friends then she'll likely either assume that she knows you or that you the two of you probably have some things in common. This is simple enough to do; just bring up one of your friends' friend lists and check out a few of the girls' pages. If she's connected to you by more than a couple of people then you're good to go. You may well stumble across some people that you've met before but forgot about in this way too.
Once someone has accepted your friend request it's only a matter of time until they pop up on the instant messenger list, and then you can work your magic. Alternatively you could try sending a message along with a friend request and starting up a little conversation that way. Something along the lines of “Hey we went to high school together! Remember me?” or “Hey you're gorgeous =)” is probably all you'll need to incite at least some sort of response. Personally I'd recommend just waiting on the instant messenger in most cases though, as an instant messaging conversation is much more akin to a real life conversation than an email back-and-forth, and if your email style messaging sputters out and you wind up sending a message with no response you might have done too much damage to recover once the instant messaging situation arises; you would probably rather have the girl develop an opinion of you via real time conversation.
Familiarize yourself with the instant messenger. In the old days, social networking sites were all about messages and comments and all communication went through in a rather delayed manner. Facebook has continually evolved and now you can communicate instantly in chat room fashion with any of your friends who are online. Instant messaging makes being flirty and charming a whole lot easier and a lot less stressful than trying to bring your “A game” in face to face situations. Having time to think about what you want to say really allows you to bring out your full personality and sense of humor without having to worry about your wit being as quick as possible. When you first catch a girl on the instant messenger you'll probably want to kick it off with some platonic “what you up to?” type small talk. Once a mini-conversation has been established you can essentially take it in two directions which can both work: try to increase the depth of the conversation by letting the girl know more about yourself and asking her questions, or turn into an overtly flirty goofball. A combination of the two isn't a bad idea either. Some girls will be very receptive to cheesiness and think its sweet while others will be more interested in a more mature conversation; play it by ear. Don't jump the gun on asking for a girl's number; if you ask too soon you'll send a bad message. Generally you're going to want to wait for a girl to start being quite receptive to your advances before you try to get the digits.
Facebook offers a very casual social neighborhood environment that really takes a lot of the pressure off of forming internet relationships in comparison to dating sites. Consider it the perfect way to find a girl to “hang out” with as opposed to finding a girl to “go on a date” with. The best thing about it is that getting rejected over Facebook is pretty much laughable as you can move directly on to the other girl who just became your friend and spit the exact same line to her as you did to the girl who just rejected you (kind of hard to do at a bar). Get with the times and have some fun; Facebook awaits.