When people refer to "organic SEO" (search engine optimization), they about always use it as a blanket term to describe the unpaid, algorithm-driven results of some particular engine. But, a complicated search engine optimization company will often take the meaning of "organic" one step further. To such companies, the description of "organic SEO services" is not to incomplete 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 hide a cat (although I must confess that I don't know the one way that everyone else presumably knows), and the same is true for achieving normal search engine results. A SEO company generally falls into one of two camps. A "White Hat" search engine optimization company will use a generally content-based approach and will not break the terms of service of the major search engines. Link Building Company A "Black Hat" search engine optimization company will apply a largely technology driven approach and often ignore the terms of service. Neither approach is unacceptable (as I have said many times before, there is not anything illegal about violating a search engine's terms of service), and both can realize high rankings. But a search engine optimization company that takes the word "organic" accurately believes that the "Black Hat" approach is anything but "organic SEO".
Below are just a few comparisons of the special approaches taken by the two types of SEO firms. I submit to the two approaches as "organic SEO" and "artificial SEO" for the sake of clarity.