Why using uppercases in database columns / tables? It is highly recommended not doing that and if you want to seperate words you should use an underscore '_'
Code:
<?php
$query = mysql_query("SELECT AppID, UserName FROM staffapps ORDER BY AppID DESC") or die(mysql_error());
while($row = mysql_fetch_assoc($query)) {
echo '<a style="color: black;text-decoration: none;" href="viewapplication?id='.$row['AppID'].'">'.$row['UserName'].'</a><br />';
}
?>
This should return a list of links to the application. (Assumed your database structure is correct)