Cannot install core in the root of site
When you download and install Drupal, it will install inside the directory that contains the files. So if you download Drupal into public_html and the directory with the Drupal files is called "drupal" it's going to install in public_html/drupal. Depending on what server you are running, you will need to identify the new webroot. In this example, you would change the webroot from public_html to public_html/drupal. The folks over at StackOverflow or your hosting company should be able to help you with your specific server settings for your new Drupal root.

Alternatively, you may be able to copy all the contents from the unzipped folder containing all the Drupal files into the webroot, but I've never tried that because I like keeping Drupal in it's own root.

Also, check sites/default/settings.php and see what it says for your Drupal root.