Link farming is the technique of having a network of sites link to each other in an effort to boost the over all rankings for the site.

While Webopedia gives a flat definition of link farming with a negative connotation, Wikipedia I think gets it right in mentioning that the technique was exploited to milk the algorithms of search engines such that Google that primarily take the links on a page into equation.

The link farming definition from Wikipedia

On the World Wide Web, a link farm is any group of web sites that all hyperlink to every other page in the group. Although some link farms can be created by hand, most are created through automated programs and services. A link farm is a form of spamming the index of a search engine (sometimes called spamdexing or spamexing). Other link exchange systems are designed to allow individual websites to selectively exchange links with other relevant websites and are not considered a form of spamdexing.

Advantages of Link Farming

Applied in moderation, link farming ( and I am not referring to the derogatory term that this has come to represent ) is a good technique to deploy to provide readers links to articles from your own site. This makes sense with any content but in particular for events that occur successively as in news follow-ups where you might consider linking to a previous article(s) on the same topic.

Considering that you would have expended quite some effort in covering the links for the previous article, the link will provide an easy checkpoint for your reader and also bring in the advantages of page ranking improvement.

And in presenting the advantages i am not referring to any secret sauce for SEO but plain logic since what is good for the readers should translate to good SEO.

Disadvantages

As with other tactics, automated link farming that seeks to pump in links to create a spaghetti of connections will ring a tocsin. It is this automated term that has come to be so dreaded in the SEO world. Hence you could call the positive technique organic link farming as well.