An organic keyword is the search query entered in, where a target website shows up in the search results.
In the above example, https://ahrefs.com/backlink-checker has a organic keyword "backlink checker".
The fastest way to check this for any target website or URL is via Site Explorer > Overview, which shows the total number of keywords that the target website ranks for in the top 100 positions of google SERPs:
Various reports also have a SERP button which you can use to check the actual SERP where the target website was found to rank:
Tips about how total organic keywords are calculated
If the target website ranks for the same keyword across different locations in “All locations” mode, it is still counted as only one keyword.
By default, both rankings in normal Organic results as well as SERP features are counted.
The reported number of organic keywords is almost always smaller than the actual number of keywords you're ranking for. This is because our database may not contain certain keywords that your target website may rank for.
Ahrefs updates its keyword database with newly-discovered search queries every 1-2 months. During this time, new keywords will be added, and unpopular keywords will be removed. Announcements about keyword database updates can be found in our Product blog.