Anatomy of a Link
The most common analogy of a link is to liken it to a vote. If page A links to page B then page A is suggesting page B is a good page, page A likes page B. The only slight complication is the nofollow tag. In a nutshell the nofollow tag allows links to be added to a page that create a click-able link to another page without the link passing a vote. These are used for text link adverts for example where they will send traffic to another page without passing a vote.
I'm not going to go into any more detail on the nofollow tag here but I'd strongly advise reading the WikiPedia page for an in depth explanation. In short if you want to link from a page on your site to a page on another site without inferring any kind of endorsement use the nofollow tag.
Why BackLinks Are Important
Having established links are like votes it's not too hard to understand why links are important. Just like a politician wins an election by securing more votes than other candidates, web pages secure the first position in a search result by securing more votes than other pages. However, unlike a political election where all votes are equal, where search engines are concerned this isn't the case, and this is where most of the confusion arises.
Why BackLinks Are Not All Equal
In order to explain this further I'm going to stick with the voting analogy and expand on it. When thinking about links as votes it's useful to consider the term block vote or block voting. Most commonly associated with unions a block vote infers upon one person the voting rights of all members of an organisation. Let's say there are 4 people in a room each representing one group of people.
A proposal is put before the group and Mr Smith, Mr Jones and Ms Samson all vote in favour with Mrs Collins voting against. On the face of it that's 3 in favour 1 against and the proposal would be carried. However, Mrs Collins represents a group of 5000 people, 500 more than the number represented by Mr Smith, Mr Jones and Ms Samson so in fact, the motion isn't carried, it's defeated by 5000 votes to 4500.
This is pretty much the same as the way links work. Simply put links carry different voting rights and it's perfectly possible for a page with only 3 or 4 links to rank higher than a page with several hundred or even several thousand links. It's this aspect of link building that's so badly misunderstood and the reason there's a whole industry offering to build links on the basis of X number of links for Y number of dollars.