What are organic search results?
Organic search results are search results that appear after a search that are not sponsored or paid for by a third party. They are usually determined by the search engine algorithm's evaluation that this result is the best matched to answer your search query.
A typical search engine result page is shown after any search. The top few results are usually advertisements placed by PPC campaigns. After the top adverts the next ten or so results are organic - matched to your query by the search engines algorithm with no third party payment. A few ads reside at the end.
Here's the search results page for a search for "seo software" on Duck Duck Go with the paid and organic results clearly marked: