Currently Google, Yahoo and Bing share just about all search traffic on the internet and as we know Google have most of the market share. Perhaps as high as 80%. Before Google arrived on the scene search engine results contained a lot of irrelevant results, illicit advertising and spam. Search Engine optimisers had discovered methods of abusing search engines and manipulating their results. Google changed this this by setting up a system whereby a website would be listed in accordance to how many other websites linked to it. A link to a website is considered a vote for that website.
Having said that, there are many factors used to determine how much weight a link has, not every link is treated equal. A link from one site may be worth half a vote while another could be worth 4 votes. Links should be relevant ie: a link from a sports related site would be worth more to your website if it also has a sports theme. Also link placement is important, a link that appears higher in a page is given more weight than one at the bottom of a page. Link age is a factor. A link may not be considered a vote by Google until it has appeared for a certain amount of time on a page (This helps reduce spam). Another factor is called trust rank. Certain types of pages always appear near the top of search results, for example wikipedia. These types of websites are highly favored by Google. A link from these types of websites can be worth a lot of votes.
It is possible to skip the queue and have our web pages placed at the top of search engine listings by using pay per click services such as Google adwords. These service are advantageous because we can tailor the copy and headings that appear in the listings as well as targeting keywords precisely. However the disadvantage is that these services can become very expensive. Especially for competitive search terms.