There is probably an error with the mysql, but we can't tell, because there is no else {} statement on the if { } statement, and nothing to show an error at the end of the query, try this, and find the error:
<html>
<body>
<form method="post" action="welcome.php">
Change the Welcome message :
<br>
<br>
<textarea name="welcome" cols="60" rows="10"></textarea>
<br>
<br>
<input type= "submit" name="welcomebtn" value= "Add your welcome message" >
</form>
<?php
include ("constants.inc");
$welcome = $_POST['welcome'];
$connection = mysql_connect($GLOBALS['host'],$GLOBALS['user'],$GLOBALS['pass']) or die ("couldn't connect to server");
$db = mysql_select_db ($GLOBALS['db_name'], $connection) or die ("Unable to select the database");
$query = "INSERT INTO settings (welcome) VALUES ('$welcome')";
$result = mysql_query($query) or print(mysql_error()."<br />");
if ($result) {echo ("Your welcome message has been updated" );}
else { echo ("Sorry, read the error above to find out what is wrong.");}
mysql_close();
echo ('got here');
?>
</body>
</html>