You must be being linked to from some other website, to that website then.
What Google does is this:
Search for text on web pages. Rank pages with the search terms you use higher, if those terms occur more frequently onthose pages.
Then, it looks for links to those pages. So if lots of people use the text "complete idiot" when they link to the web page showing G. W. Bush's bio, the bio page will be highly ranked if you search for "complete idiot". (I use that example because it's real, or was.)
So if your name is no longer anywhere to be found on the organisation's website, perhaps there are other links around the web that say (for example), "See Khandro's page" and link to the organisation's website, the organisation's site will still appear if your search for your name.
If this is the case, you could try contacting Google, but I would doubt anything will be done. The easiest thing to do is to remove the links of your name to the organisation's website.
Can you give us a link to look at, to see if that would help in giving better answers?