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. 

It is best to select around 5-10 key words or phrases you would expect your users to be searching for, or do some research on finding out what exactly users were searching for previously before finding your site. Make sure these keywords are shown in the title and body text of your site on 1 or more pages. Add useful content that users may wish to know or read that include these keywords and phrases. Contact other related sites and have them link to relevant pages so you can share content.  Keep your site clean and tidy, and make it easy for users to navigate, avoid using flash or JavaScript only navigation methods. Over time, keep adding new fresh content and your natural search results will improve, and best of all, these results are free! So are well worth the effort.