That's really the number one question, and for many businesses the answer is as different as they are. I've been developing websites ever since there was a web and in the beginning there wasn't enough of a track record to really prove what the Internet could do. Microsoft didn't understand it yet and Google wasn't even a twinkle in anyones eye. It took many years for business to understand that this new Internet thing wasn't just a passing fad, but instead would become the integral business / social tool that it is today. I can recall having many a meeting with prospective clients trying to convince them that the "World Wide Web" was a valid way to expose their business, create new opportunities and to streamline existing customer relationships.
Well thankfully the days of convincing people about the Internet is over but now we have different challenges. The Internet is a living organism in every sense, it grows, it thinks, it is creative and it is unpredictable. The next Google or Facebook or YouTube might happen tomorrow it might happen in a year and it will be something that hasn't yet been conceived by the majority of us, but when it does get "birthed" it will make sense and we'll think "why didn't I think of that?" and "What did we do before this?"
This takes me back to my original point, and yes I do have one. "What makes a successful website?" Ideally, the simple answer is "Does the website achieve the goal(s) it was created to achieve?". Most website owners don't know the answer to this, in fact they often don't know what the goal was in the first place. Do anything without a plan and you'll probably find a lot of things on your journey of discovery, but you won't reach your destination anytime quick.
So be clear in what you want from your website and set some measurable key performance indicators. Do you want to makes sales or generate leads via the internet? if so you need to track and measure your sites effectiveness in these areas, quantify the results and review them. make sure your site has the tools to do this and ensure you can adjust your strategy in the event you aren't achieving the results you want.
This goes for anything you want from your site:
Plan > Measure > Review > Adapt - and the cycle continues.
Become an expert on YOUR Internet strategy. Keep up to date, because as I mentioned before the Internet evolves, your business needs to evolve with it.