When people refer to "organic SEO" (search engine optimization), they almost always use it as a blanket term to describe the unpaid, algorithm-driven results of any particular engine. However, a sophisticated search engine optimization company will often take the meaning of "organic" one step further. To such companies, the "organic SEO" is not limited to what shows up in the "natural" search engine results—it includes the methodologies used to achieve such rankings.
There's more than one way to skin a cat (although I must admit that I don't know the one way that everyone else presumably knows), and the same is true for achieving natural search engine results.
A search engine optimization company usually falls into one of two camps. A "White Hat" search engine optimization company will use a largely content-based approach and will not violate the terms of service of the major search engines. A "Black Hat" search engine optimization company will use a largely technology driven approach and often ignore the terms of service.
Neither approach is invalid (as I have said many times before, there is nothing illegal about violating a search engine's terms of service), and both can achieve high rankings. But a search engine optimization company that takes the word "organic" literally believes that the "Black Hat" approach is anything but "organic SEO."
Merriam Webster defines organic, in part, as "having the characteristics of an organism: developing in the manner of a living plant or animal." To a search engine optimization company, this definition accurately describes the approach taken to achieve long-lasting results in the "natural" section of search engines.
Below are just a few comparisons of the different approaches taken by the two types of SEO firms. For the sake of clarity, I refer to the two approaches as "organic SEO" and "artificial SEO."
Content vs. Technical Loopholes
There's an "old" saying in the SEO industry that "content is king." This is not necessarily true. In my experience, good content is king. Study after study has shown that when people use search engines, they are primarily seeking one thing: information. They are not seeking to be impressed by fancy flash sites. They are not looking for a virtual piece of art.