Links are the votes of endorsement that power Google's PageRank algorithm. Links are the single biggest indicator to Google that a web page and a domain are authoritative, and therefore should appear highly in search results for pages with content relevant to a searcher's query.
In order to improve their organic search traffic, many websites have paid to have links built pointing to their site. This is a clear attempt to manipulate search results and violates Google's webmaster guidelines. A paid link is not an authentic endorsement of a site, similar to an advertorial in a newspaper. To let Google know that a link is in fact a paid-sponsorship rather than a natural endorsement, links can be "no-followed" which let's Google know that the link should not pass PageRank.
Meet Matt Cutts. He is the head of Google's Web Spam Team. If you buy links or engage in other tactics designed to manipulate Google's algorithm he is likely to show up and Google is likely to penalize the website.
So what actually happens if you do buy links and engage in engine-manipulative behavior? Google penalizes or even bans your website from search results. It can be incredibly difficult or impossible to fix the damage - many times sites have to buy an entirely new domain and start from scratch with their link profiles.
Below are some major brands who have learned the hard way.
Way back in 2006, BMW was one of the first major manipulation scandals to be exposed.
In 2011 the New York Times covered JC Penney's link buying scandal and the penalties that resulted.
In 2012 a top-notch SEO firm called iAcquire was delisted for link buying allegations.
In 2013 the highly regarded Mozilla Foundation was penalized over user-generated content.
In a 2013 highly publicized move, Rap Genius was penalized for a link acquisition scheme.
In 2013 the BBC was hit with a manual penalty.
And as recently as 24 hour ago it appears Matt Cutts is on the trail of T-Mobile buying links.
What does Matt Cutts think about this?
The moral of the story is this: Do not buy links. Do not sell links. Do not join a "link network". Do not join a "guest blog network". Do not engage in any type of automated link creation. Do not try to manipulate Google's algorithm.
Create great content. Reach out to people you know to share your great content. And under no circumstances, never, ever buy SEO services from a place like Fiverr - it will result in a penalty for the site.