“search results” template

could anyone here explain me how searching works?

After clicking the search button I noticed that I get directed to –> SomeDomain/index.php?s=STRING&submit=Search

Bu some small debugging I also noticed that from what ever page template I do the search…the results’ page is always a template of the index.php, How can I change that and use my own template for the results….let’s say Search.php ??

Any ideas?