Quite a few things wrong; instead of listing each fix it was a lot easier for me to do a rewrite.
if($user_logname && !$submit){
$Query="select user_name from $TableName order by user_name";
$Result=mysql_db_query($DBName,$Query,$Link);
$Result_num=mysql_num_rows($Result);
if($Result_num>0){
echo '<form method="post" action="email.php">');
echo '<select size="1" name='emails'>';
for($i=0;$i<$Result_num;++$i){
$column=mysql_fetch_array($Result);
echo "<option>$column['user_name']";
}
echo '</select>';
echo '<p><input type="submit" value="Submit" name="submit"><input type="reset" value="Reset" name="B2"></p>';
echo '</form>';
}
}