web folder creation and permission

Before giving a definite answer, some information would be useful:
Is the script to be run by a web server through some GET/POST request (and if so does the web server run as root user?), or by a user logging in to your system or by some other method?
Is the username you are talking about a real "unix" user. i.e. a username listed in /etc/passwd who can run processes and own files and directories or just a user known to the web server (they may not coincide)?

Supposing that the user is a unix user, and the operation has to be done by a web server, then either the web server runs as root (this is not usually the case), so that you can simply do something like

su -c "mkdir $dir" $user

or you have to acquire the user's permissions through some authentication mechanism like ssh or sudo; with sudo command, for example (see man sudo), you can give limited permissions to some particular user to execute only a limited amount of commands as another user, without need to know that user's password.

yes script run with GET/POST request and as root user.

actually my task is simple - first a user come to my web site and he/she registered there.

after successful registration i want run a  shell script, that create a folder with a unique name like-"/9087" on the root of my web server's space.

and set the username and password for access folder by using FTP same the user enter on my site at the time of registration.

now - user simply enter my site host name, user name and password(that is same as registration on site) on any FTP S/W and access only the assign folder like"/9087". and upload some date on it.

i use LAMP for this.

Well, now I understand that your purpose is to create an ftp account for new users that register in your web site, pointing at the user's web root directory and sharing the same password with the web server. As far as I know, this is usually accomplished through "virtual users" i.e. users which are not contained in /etc/passwd for security reason, they are associated only to a numerical Unix user id and are listed in a special file known by the ftp server. If you are using proftpd ftp server you should have in /etc/proftpd.conf at least these lines

# virtual user/group files
AuthUserFile /etc/ftpd.passwd
AuthGroupFile /etc/ftpd.group
# restrict every user to its own home dir
DefaultRoot ~

Then, when a user registers, your web server should execute a script more or less like the attached one, where the first argument is the numerical user id, the second argument is the user name and the password is given as standard input (for security reasons), of course WEBROOTDIR should point to the root of the web server's web space.

#!/bin/sh
UID=$1
NAME=$2
WEBROOTDIR=/webserveroot
USERHOMEDIR=$WEBROOTDIR/$UID
# could be USERHOMEDIR=$WEBROOTDIR/$NAME
 
# Security test
if [ "$UID" -lt 500 ]; then
echo "Error user id $UID too low"
exit 1
fi
# Use ftpasswd utility from proftpd for creating user and setting password
# password is received from stdin
ftpasswd --passwd --file /etc/ftpd.passwd --uid $UID --name=$NAME --home $USERHOMEDIR --shell /sbin/nologin --stdin
# Create home dir
mkdir $USERHOMEDIR
chown $UID $USERHOMEDIR
                                              

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