Here is what happened, most likely.
The file you are deleting for some reason does not belong to your account. Check with "Get Info" and look in the tab "Sharing & Permission".
Either the file belongs to some other account, or you do not have "Write" permission.
Nothing alarming; for example you may have copied the file in some strange way that preserved the old permissions.
In any case the computer is asking for your administrative password. This may well be just your password (if you are admin).
Just go ahead and forget about it, there is nothing wrong.
(You may also change the permissions first for the whole directory, but you should do that from the terminal (at least this is the only way I am familiar with)).