Use chmod +s <filename> to se the suid bit. This means that when the file is executed, it runs with the permissions of the owner of the file (so chown it to root to make it run as that).
However, this can be VERY dangerous with something like a bash scripts, because a user finds a way to change it, they can easily gain a root shell. Make sure that it can't be written to by anyone except root.
Linux doesn't allow you to setuid on scripts, to do this you'd have to compile it as a program. Instead, you can use the sudoers file (/etc/sudoers), and add a line like this.
<username> ALL = NOPASSWD: /path/to/script