First, make sure you've filled out your complete job history. That's what will start you down the path of making connections with current and former co-workers. By making connections with these colleagues, you'll start building your network.
It's easy to get one of your primary connections to introduce you to one of their connections. But when you ask for a third-level connection (a colleague of a colleague's colleague) things start to get a little tricky. The more connections you make, the more likely you can get an introduction to a company where you would apply. By getting the introduction first, your resume can be moved to the top of the pile.
These days, when several hundred resumes are sent for every one position, that introduction can mean the difference between a call and your resume being overlooked.