Before your website can get listed on a search engine, the search engine must first of all know that your website exists. Search engines do this by using a web crawler.
A web crawler is a program that methodically browses through the websites on the internet and allows the search engine to index them. How a web crawler works can be crudely summarized by these steps.
1. The web crawler visits the webpages that it has previously indexed.
2. It searches those webpages for outbound links.
3. It visits those links and indexes the pages that are new.
These steps are repeated as new webpages are found.
As you can see, the best way to get the search engine to index your page is to get another webpage that is already indexed to link to yours.
The other, less recommended, way of getting your webpage indexed is to submit your webpage URL to the search engine directly. This lets the search engine know that a webpage exists and needs to be crawled. However, this does not guarantee that your webpage will be crawled.
Here are some of the URL submission pages for the more common search engines.
Google - http://www.google.com/submityourcontent
Bing & Yahoo! - http://www.bing.com/toolbox/webmaster
If you want to submit your webpage URL to the search engines directly, it is recommended that you submit it to Google, Bing, and Yahoo!. This is because online sources say that these search engines sell their search results to the other smaller search engines.
Remember, when building a website, CONTENT IS KING. If you website has good content and is updated frequently with fresh material, people will come and naturally link your webpages.