The Value of the Link
It used to be that you could tell search engines what you wanted to rank for. With "meta keyword" tags, you could enumerate some terms and phrases, and if you liberally sprinkled them across your page, you had a good chance of ranking for those terms. My, how the times have changed.
Google's patented PageRank algorithm (named after Larry Page, not web "Page") sought to refine this by, among other factors, using links to a page as "votes" for that page. The theory was that if a page is linked to more, it was probably more important. Also, links from important places are more valuable than links from less important places.
A link has three parts: The source it comes from, the destination it goes to, and the anchor text, or visible text of the link. Google takes all three of these into consideration, as well as the age of the link.
As Google rose to dominate the search engines, a new link economy emerged. People of questionable ethical standards would apply techniques such as spamming blog comments with links back to their site. Google and these spammers are in a constant struggle, not unlike the relationship between antivirus software and virus creators.
Today, link spamming sometimes works, but most blogs use nofollow tags to prevent Google from counting it as an important link. Additionally, Google actively penalizes sites who, in their sole discretion, are abusing the system. A penalized site may not even rank in the top three pages for their own name.
In general, to rank highly for a competitive term, you would need more and/or better links than your competition. The process of acquiring these is known as link building.