A network is basically a big web of contacts in your professional field or area of interests. Your network may consist of current of former or current acquaintances, colleagues, or employers with skills in certain areas of interest to you. They might be high school teachers who have friends in high places, or a family friend who works at a company that you'd like to land a job at. There is virtually no limit to who can be a potentially useful contact.
Keep your options open and make friends, not enemies, wherever you go. I have heard of people sitting next to useful contacts on the train, or on a bus—there's no way to know when a casual encounter with an acquaintance will turn in to an exchange of business cards.
Keep in mind that an "in" at a company is one way of moving past your competition and gaining an advantage over them. If one of your personal contacts is at a low level within an organization, he or she may not have much influence over helping you to secure a job. Still, he will be able to pass your resume on to someone at a higher level who may have more authority and the power to offer you a position.
In order to make a network work for you, you have got to be persistent. Success in the job market is all about timing, and it may not be the case that a contact will have a job for you the moment you inquire about one. But you should make a point to stay in contact—make a phone call or send an email every few weeks to check in. That way, you will be strongly in the contact's mind if a job does open up.
Of course, you should be judicious in terms of deciding which contacts to follow up with and which contacts are not likely to lead anywhere. If you've put in multiple phone calls or send several emails to someone who you thought might be able to help you, but they haven't responded, it may not be worth your while to continue pursuing them. In this situation it's better to focus your time on more responsive contacts who reply to your correspondence and inquiries.
The key to great networking is to build relationships over time. Don't give your contacts the impression that all you're out to do is secure a job. Instead, make them feel as though building a friendship is your primary interest-which, of course, is what you should be doing. If the contacts feel a connection with you, that will make them all the more likely to help you in your job search.