I am not sure if you are asking if your site should be famous or how it can become famous. Obviously, when your site is popular you will reach more and more people. This makes the site more profitable, so that you can make money from advertising or selling content. As for how your site becomes famous, you do so by being original and having constantly changing content. When people are talking about you, you'll become famous.
First I think it fully depends on what kind of site you are running. Blogs, for example, find success by targeting a specific readership group and (like Akannan said) updating frequently. Amazon found its success by being one of the first of its kind and then growing so large that it was possible to find virtually anything on there and compare prices or buy used. Facebook likely found its success by targeting college students first, then strategically expanding as those students entered the working world.
Because your question is so general - I'd generalize by saying a "famous" website (one has frequent and numerous visits) probably does one of the following really well: does something that no one else has done yet or takes a new twist on an old idea and improves it; provides information that people want and need; entertains; offers something for free; offers something at a cost that people cannot obtain anywhere else.
There are many internet paths that one can pursue in order for one's site to be famous. The term "famous" for the internet is a bit of a challenge. The concept of the world wide web and the amazing amount of speed it presents helps to make the argument that being "famous" is actually transitory. When one concept gains fame on the internet, it runs its course and another element will take over. For instance, we can gauge this on the internet with traffic and activity. Last summer was a great example of this. In July, the student protests over the Presidential election in Iran took center stage. Internet traffic was a buzz with tweets and posts on Facebook. This was dislodged from activity by the shooting at the National Holocaust Museum in Washington. The student protests in Iran took second stage as people were seeking to find out details about the shooter and his background, including his writings. This event took receded in interest with the death of Farah Fawcett, but this was short lived as about eight hours later, the internet traffic was sprawling with news of the death of the King of Pop Music, Michael Jackson. In the end, this news cycle demonstrated that internet attention and traffic is generated by whatever is the latest "buzz." Holding people's interest is a challenge in such a venue. I think that if one wants to attract many people to their site, then being able to predict and read the mood of people, especially young ones, will be critical. Additionally, your site should be updated frequently with relevant information to ensure that traffic does not bypass it out of disinterest.