Update: I've expanded this answer into a longer article, Overcoming The Network Effect Barrier.
Even though this answer is for social networks, a lot of this advice could be applied to any business or website as well.
Be The Best At In At Least One Area - Don't be just another clone. There are enough of those. What is your unique selling proposition? Twitter is the best at short messages. Most other social sites now have some form of the feed, but Twitter remains on top for short messages. Facebook has the best (largest) network. YouTube delivers video best. Have at least one thing that you can hang your hat on, one core feature that you will continue to improve and not let anyone do better than you.
Use Incentives - Give something away, hold a contest, offer your service free while in beta, give points for using the network, make the points redeemable.
Low Barrier To Entry - Sign up should be painless with very few fields that are mandatory. Importing or finding existing contacts should be made simple. If you can piggyback on existing solutions like Facebook Connect, OpenID, OAuth, or even Gravatar, it makes it easier. Depending on the type of social network site, you may be able to skip requiring registration all together.
Make It Sharable - Give reason for people who aren't members to visit. Facebook, Flickr, SmugMug all have publicly sharable galleries; Tagged emails you when friends are using it; Digg, Twitter, and countless other social sites have public interfaces.
Make It Sticky - Why should users keep coming back? How are they being engaged? It might sound silly, but Facebook probably wouldn't be as far along as it is today without Farmville and other social apps.
Easy On The Ads - There are other ways to fund your social network without deluging your users with ads. Micro-transactions are a great start and fund a lot of social gaming companies. Freemium is another model. Find ways to integrate businesses as a core part of your network and then charge businesses for premium features like analytics or promotions. If you do have advertising, make it fun, engaging, and memorable. Note that flashing banners are none of those things.
Enable, Empower, Enlighten - Make your users feel good about themselves while they're using your social network. Every experience should be a good one. The focus shouldn't be your network, it should be the awesome things that it enables your users to be able to do and learn. Your network is just the tool. Don't lose that perspective.
Evolve - What your network is in the beginning shouldn't be what it is in the end. Facebook started out with requiring a Harvard email address, then it spread to other schools, then they opened it up to everyone, then they added a social layer and now email. Twitter adopted hashtags and @replies into their core product. Both have an API. Keep an eye out for how people are using your network in unexpected ways.
Listen To Your Users It doesn't always mean that you need to implement every feature they ask for, but you should at least have an open door policy. Facebook has both listened and ignored user protests. They rolled back Beacon, but kept the News Feed. Both were heavily protested.
Be Transparent - Unannounced or wholesale changes without user input don't always go over very well. (See Digg). Users like to be kept informed on what is going on. Maintain a blog. Solicit input. Plan ahead. Stick to your core principles and purpose.