You need to access your form/cookie variables through $GET, $POST, and $_COOKIE arrays.
So if your URL is like this:
whatever.php?foo=qwerty&bar=asdf
Previously, with register_globals on, doing this:
echo $foo."<br>".$bar;
Should now be this:
echo $GET['foo']."<br>".$GET['bar'];
Both of which will output:
qwerty
asdf
If it is a POST form instead of a GET form, use $POST[] instead of $GET[]. Likewise, use $_COOKIE[] for variables set in cookies.
The reason it is better to have register_globals off is because if you have something like:
if ($logged_in) do_something();
someone might use a URL like http://www.example.com/whatever.php?logged_in=1
and possibly bypass your check, if register_globals is on.
In general, it is bad (from a security standpoint) to give the end user so much control over the internal state of your program as register_globals does.
Note that in older versions of PHP (before 4.1.0 I think) you need to use $HTTP_GET_VARS[] $HTTP_POST_VARS[] and $HTTP_COOKIE_VARS[] instead of $GET[] $POST[] $_COOKIE.