Simple Ajax Jquery script- How can I get information for each of the rows in the table?

The old MySQL extension mysql is outdated, better use mysqli or PDO!

mysql_fetch_row() returns only 1 row! You have to put it into a loop, for example:

$data = array();
while ( $row = mysql_fetch_row($result) )
{
  $data[] = $row;
}
echo json_encode( $data );

You also have to change the JavaScript:

$.ajax({                                      
  url: 'api.php', data: "", dataType: 'json',  success: function(rows)        
  {
    for (var i in rows)
    {
      var row = rows[i];          

      var id = row[0];
      var vname = row[1];
      $('#output').append("<b>id: </b>"+id+"<b> name: </b>"+vname)
                  .append("<hr />");
    } 
  } 
});

By the way I would recommend you to use mysql_fetch_assoc() because it makes your code more flexible and cleaner.