Like many industries, search engine marketing is rife with jargon.  One of the terms that you'll hear frequently is "Natural Search Engine Optimisation" and it can have a couple of different meanings depending on who you speak with.

I've heard this term used as another version of "organic seo" or the optimisation of web sites for the natural listings in search engines such as Google.  These are the "free" search results that are provided by search engines and usually appear on the left hand side of the search engine results page (SERP) - paid listings typically appear on the right side of the major engines.

The order of these results are determined by a complex computer program (also known as an Algorithm) that determines the relevance of web site pages to a particular search query.  No one is paying for these results, but problems with a company's web site can prevent it from ranking highly for terms that it should.  Search engine optimisation helps these sites better represent themselves in the search engines by making their sites more efficient, more relevant and more "search engine friendly".

The other way that "Natural Search Engine Optimisation" is sometimes used is to describe the way in which SEO is performed. 

Search engines are in a constant battle to prevent people from "gaming" the SERPs.  Many people are perpetually trying to find short-cuts and loop holes to the search engine's algorithms.  Others just simply over do it.  They take a concept such as the importance of building the reputation of a site through "off page" techniques like link building and go too far.  They create hundreds or thousands of links very quickly for a company's website.  Or perhaps they over-use the keywords that are being targeted in the content of the site.   These concepts used in moderation are fine - but too much is considered "search engine spam" and could get a site penalised or even banned from a search engine.  The search engine's are actively looking for signs of "over-engineering" the optimisation process.

In this context, the term "Natural Search Engine Optimisation" is about optimising and promoting a web site to get the desired result, but by following a process that is "natural" and not over-engineered.  This involves removing any impediments to the site structure that could prevent it from gaining rankings for particular terms, adding real, relevant content that has value to the company's website visitors, and promoting the site in a way that makes sense - that would be done naturally, if the company had the time and/or knowledge to do so.  This will always prove to be the most successful method - long-term - in promoting a web site; anything else, should be considered with caution.