The network effect is a phenomenon where the value of something to an individual increases with the number of people who chose the same thing. This situation typically occurs in networks, where many people are connected in some way. The technical term for this in economics is network externality. In economics, an externality is where one person's choice affects another person's choice.
When telephones were first invented, demand was low because there weren't many people who had a phone, so there weren't many people you could call - limiting the value in getting a phone. The more people who got phones, the more value there was for other people in getting a phone (as there were more people they could call), and when they got a phone it made it even more valuable for the next person to get a phone. The number of subscribers grew exponentially. The same thing happened when fax machines were introduced.
If your friends all use AOL's IM, it makes more sense for you to use it too, rather than competitors such as MSN's Messenger. The more people there are who use AIM, the more new people want to use AIM too. That's the key characteristic of the network effect - the value to joining a network increases with the number of people in the network.
Note: now there's software that can let you use multiple IM systems, so there's less pressure to pick one and stay with it.
eBay is an online marketplace. The more people sell on eBay, the more buyers there will be and the more incentive there is for new sellers to use eBay.
Computer Operating Systems
The more people that choose to use Microsoft Windows, the more valuable it becomes for the next person to choose Microsoft Windows, as this person will be able to exchange files with the many existing users without difficulty. As the number of users of Microsoft Windows increases, it is more valuable for software developers to write software for Windows, as there are more potential customers. The availability of more software makes Windows yet more valuable for new computer buyers - and so on, and so on. That's why Bill Gates is so rich!
PayPal's story is similar to eBay's. The more people who use PayPal, the more other people have an incentive to use it.
FaceBook and LinkedIn are social networks, and - being networks - they exhibit the network effect. If most of your friends are on FaceBook, it makes sense for you to use FaceBook too, instead of a competing social networking site.
Social networks sometimes need a kick-start to build a critical mass of users. Many new sites use Facebook Connect, allowing people to use their Facebook login to connect to new sites.
Larger Networks Offer Greater Customer Value
In many industries where one company is highly profitable, new competitors will enter to try to get some of those profits for themselves. So long as the new entrants offer customers the same value as the incumbent, they should take some market share. For example, if one airline flies from London to Turin, another airline can also offer flights from London to Turin. From the point of view of the customer, the offering is the same.
But network effect businesses are different, because new entrants cannot offer the same value to the customer as existing businesses. If you want to sell you grandmother's antique table lamp, you will choose to sell it where you can get the best price. You will get a better price if you have access to more buyers. That gives you an incentive to list the product on eBay, instead of a competing online auction service. A new competitor just cannot offer you access to the same number of potential buyers. All sellers reach the same conclusion, and the dominant company just keeps getting bigger and bigger.
Network Effect Businesses Grow Rapidly and Create Monopolies
Networks tend to grow rapidly. They also tend to create monopolies, as the largest network is the one that is the most attractive to join, so it keeps getting bigger and bigger (the snowball effect).
Shapiro and Varian (1999): Chapter 7: Networks and Positive Feedback
Katz and Shapiro (1985):