There are several search engines out there. The search engines (like Yahoo® and Google®) send out what are called "spiders." The spiders can follow links to web sites. They are constantly doing this. When they arrive on a page, they tell the search engine what is on the page so the search engine can index it and make it searchable. It's similar to an index in a book.
From the text(not the images) on a web page, the search engine decides what the page is about. In addition to this, it checks the sites that have links to the page. If there are lots of links to the web page or it's linked to from some big sites, the search engine decides that the link must be valuable and thus pushes it up in the search results. This is called popularity. Search engines have found that a page is more likely to have good information and be relevant if it has greater popularity.
Aside from search engines, which are like a books' index, there are "directories" (like LookSmart®), which are like phone books. Instead of sending out spiders, directories usually count on you going to them and submitting your website. Directories often have real people that decide whether to include your submission or not.
Most search engines and directories offer some form of paid service where you pay to make sure you're included in the search results.
In my opinion, the best strategy for showing up in the search results is to get listed in a directory, submit your site to search engines, share links with other websites, and especially: create meaningful content. Put valuable resources on your site and search engines will reward you for it. Share your knowledge. The search engines will find your site in the directory and rank you highly for popularity.