Your site also does not have a single overall ranking on search engines, but many different rankings, one each for a different keyword related to your site.
The quickest way to look up what your search ranking for a particular keyword is to run a search for that keyword in a browser window using incognito mode. If you don’t use incognito mode, the search rankings and results will likely be changed based on your search and browsing history.
If you want to see all the organic keywords you are ranking for, the best and most accurate way is to use the free tools provided by the major search engines. Using Google Search Console or Bing Webmaster Tools, you to look up your average ranking for all the keywords bringing traffic to your site, along with lots of other valuable data.
Like a lot of things in SEO, this is a more complicated question than it may initially appear. To answer it we need to first define what you mean by ranking. Your site may get no organic traffic even if you technically appear (or “rank”) somewhere in the search results. About 95% of search traffic goes to the sites on the first page of search results (usually the first 10 results listed), so if you are the 200th site listed it is highly unlikely that anyone is going to go through 20 pages of search results to find it.
Even if you are only looking at a single keyword, there is not a definitive rank where your site will always appear in the search results. This is because modern search engines use factors like an individual’s search history and location to customize the rankings to try better match what they think the searcher is looking for. You can block this personalization and get an unmodified ranking by running your search in a browser window using incognito mode. If you use Search Console, you can get an average ranking for a keyword, which takes into account all the personalization changes the search engine makes when showing the results for that keyword.
Search Console also provides an overall average rank for entire website, but this can be a misleading metric to measure changes in rankings or traffic. This is because it is entirely possible for the average ranking to increase even when you are getting less traffic, or decrease when you are getting more organic traffic. How is can this be? The overall rank is an average of the rankings for all the keywords you rank for. You can increase site traffic by not only ranking higher for keywords, but ranking for a larger number of keywords. If you add a lot of new keywords, you have more sources of traffic, but they may have lower rankings than previous keywords you ranked for and drive down the average rank.
If you are wanting to look up all the keywords that any site ranks for, there are services available such as SEMRush that let you do this, although most you have to pay for. The data you get from these services is based on data collected by independently of the search engines. This means that information is not as complete or reliable, although it can still be very useful when researching keywords and competitors.