When a search engine ranks your web page, it places a relevance on a number of different web page attributes, such as meta-tags, keyword density, and link popularity. The more weight your web page has compared to other pages found in a search, the higher it will rank. The objective is to return the most relevant results.
Pages that have many incoming links are likely to be more relevant than sites with fewer incoming links - these pages have a higher link popularity.
However, the story doesn't end here. Some search engine promotion services started "link farms" which basically placed your link on hundreds of other sites in exchange for them placing a link on your page, so the relevance factor disappeared.
The search engines have a way round this. They first gather the results of a search based on meta-tags, keywords, and link popularity, and then rank these results using the link popularity within this cluster of results. This means that if your site is about "web design" and you have other "web design" sites linking to you, you will get a higher ranking than if you have "pet food" or "garden equipment" sites linking to you.
Although link popularity is now used by some of the major search engines as the most important factor in ranking a web page, it is still very important to remember that the building blocks of a website are the words on the page and without these, your site would not even be ranked.
Although the link popularity factor may get the most weighting, it is only important in relation to the title, keyword and description tags, and the keyword density of the word or phrase being searched for.