Whenever the Drupal Core is updated, the default settings in the Search Box Module put the words "Search This Site:" in front of the box. To remove or change those words, use an FTP program (Control Panel or other) to access the file

modules > search > search.module

Make a back-up copy of this file on your home computer.

Use the FIND Function to find "search this site" then either delete the line shown below in bold or change the words inside the single quotes.

function search_box(&$form_state, $form_id) {
  $form[$form_id] = array(
    '#title' => t('Search this site'),
    '#type' => 'textfield',
    '#size' => 15,
    '#default_value' => '',
    '#attributes' => array('title' => t('Enter the terms you wish to search for.'))

Save the File.