The Natural Search, sometimes referred to as an Organic search, is the search results displayed which are not sponsored or paid links. They are the non-biased natural/organic results you get whose rank has been determined solely on the search keywords entered and its relevance.
Paid search on the other hand are results displayed usually above or to the right of the natural search results.
You should aim to have your site as high as possible in the natural search results before investing in paid advertising. Most companies try to focus on getting a select few keywords targeted, that is getting your site to rank high when a particular set of keywords are searched for.