Organic search results are links displayed on a SERP (Search Engine Result Page) for which the owner of the linked page did not pay the search engine.
The term "organic search result" is in contrast to "paid search result" or "sponsored link" which are ads.
There are different types of organic search results:
classic search result
search result with snippets
local search results
The place granted to organic results on a SERP vary according to search engines and according to the type of query.
In theory, the first places in organic search results for high volume queries are a big source of free traffic. In reality, that source of traffic is seldom free. Indeed, the first places are most often the result of SEO projects or services which can be costly.
Notwithstanding, organic search results are often a cheaper source of traffic and acquisition than others (SEA, display, affiliate marketing, traditionnal media, etc.).
Organic links are obtained by the means of SEO (Search Engine Optimization) which consists of "sticking" to the search engine algorithm.
An example of SERP with only organic search results:
A SERP with organic and paid search results: