A link farm is a group of websites that all hyperlink to every other site in the group.
Most link farms are created through automated programs and services and can include thousands of unrelated websites all linking to each other. Link farms are a quick, easy and relatively cheap way to build bigger link portfolios for each site involved in the link farm. However, link farms are typically viewed as a way to spam the search engines and are therefore considered a black hat tactic.
Site owners that belong to link farms are trying to manipulate the SERPs by artificially inflating the number of links a given site has. Link farms are more about quantity and not about building quality links that the search engines recommend site owners seek out.
Since the search engines look at how many inbound links a site has when determining where it will fall in the SERPs, unscrupulous site owners and spammers list their site on link farms in order to build a large link portfolio very quickly. Many of the search engines updates are focused on penalizing sites that are apart of link farms.