A website can get visitors by many different methods depending on products or services offered by a website and the publicity method employed. Websites needs to be publicized in an appropriate way so that it can reach out visitors.
In case a website needs to get visitors from search results of various search engines, then the website must be created in a search engine optimised way and should be publicized at relavent websites. The method of publicizing a website on other websites is often referred to as link building. In case you do know of another relative website, then a good place to start can be by finding websites linking to that relative website.
Yahoo Site Explorer can tell you which websites link to a given website. It can be really helpful to know which websites are providing backlink to a given website so that you can also determine where your competitor is publicizing their website and can show you a rough roadmap to follow. In case you have designed a website with subdomains, you may find other post titled Domain and SubDomain from Search Engine perspective.