..............
<form name="form1" action="selectcountry.php" method="GET">
Select your country:
<select name="country" onChange="form1.submit();">
  <option value="">Select your country</option>

  <option value="USA">United States of America</option>
  <option value="Turkey">Turkey</option>
  <option value="UK">The United Kingdom</option>
  <option value="Germany">Germany</option>
  <option value="Russia">Russia</option>
  <option value="China">China</option>
  .............
  //place here all the countries in the world :D (by typing all of them or selecting from a query)
  .............
</select>
<input type="submit" name="submit1" value="List the universities in this country">
</form>
<?
if ($_GET['country']) {
?>
<form name="form2" action="university_selected.php" method="GET">
<?
$myconnection1 = mysql_connect($db_host , $db_user , $db_password) or die ("Can not connect database:".mysql_error());
mysql_select_db( $db_name , $myconnection1 ) or die ("Can not select database:".mysql_error() );
$myquery1 = mysql_query("SELECT * FROM university WHERE country=".$_GET['country']." ORDER BY university_long_name ASC",$myconnection1) or die ("Can not select universities:".mysql_error());
  if (mysql_num_rows($myquery1)>0) {
?><select name="university_id" onChange="form2.submit();">
<?
    while ($recordset1 = mysql_fetch_array($query1)) {
    ?><option value="<? echo($recordset1["university_id"]); ?>"><? echo($recordset1["university_long_name"]); ?></option><?
    } // while ($recordset1 = mysql_fetch_array($query1)) finishes
?></select>
<input type="submit" name="submit2" value="Select this university">
</form><?
  } //if (mysql_num_rows($myquery1)>0) finishes
  else {
?>No university has been found in this country or no university has been written for this country<?
  } // else -2 line finishes
}  //if ($_GET['country']) finishes
?>

...................
You must analyze the selected university in university_selected.php file.

.................
<form name="form1 action="yourfile.php" method="POST">
................
Type your country: <input type="text" name="country" value=""><br>
Type your university: <input type="text" name="university" value=""><br>
................
</form>
.................