Can't change permission/ownership/group of external hard drive on Ubuntu

Check the filesystem type it's using first with df -T:

sys@p:~$ df -T
Filesystem    Type   1K-blocks      Used Available Use% Mounted on
ext3          ext3    19222656   1050948  17195164   6% /
tmpfs        tmpfs     1684988         0   1684988   0% /lib/init/rw
udev         tmpfs       10240        64     10176   1% /dev
tmpfs        tmpfs     1684988         0   1684988   0% /dev/shm

If it's mounted on /mnt/external for example you will see that in the far right column. You can see the filesystem type in the second column. If it's NTFS, you'll want NTFS-3G (probably already installed, if not sudo apt-get install ntfs-config then gksu ntfs-config). Linux already has FAT support for read & write although they do not support permissions.

If you want an NTFS partition mounted with ownership applied to a specific user/group, specify it in the mount switches:

mount -o uid=username,gid=groupname /dev/sdc /path/to/mount

If you change to ext3 as suggested above, you can use chown:

chown -R user *
chown -R user .

As Kim said, you'll only get Unix permissions and ownership on a Unix filesystem. ext3 is a good candidate.

If you must use this drive without reformatting, you can do it with options to the mount command that specify the owner, group, and/or read/write permissions. These options affect all files on the drive (see John T's answer for how to determine the FSTYPE):

# list files as owned by X, use numerical UID as found in /etc/passwd
$ mount -t <FSTYPE> -o uid=X /dev/?? /path/to/mount/point

# list files as owned by group Y, use numerical GID found in /etc/passwd
$ mount -t <FSTYPE> -o gid=Y /dev/?? /path/to/mount/point

# list files as accessible per umask 
#   (022 gives rwx permissions to owner, r-x permissions to everyone else)
$ mount -t <FSTYPE> -o umask=022 /dev/?? /path/to/mount/point

# combine all of the above:
$ mount -t <FSTYPE> -o uid=X,gid=Y,umask=022 /dev/?? /path/to/mount/point

I did this and it worked:

sudo umount /dev/sda3 /media/windows1
sudo umount /dev/sda5 /media/windows2

and then

sudo mount -o rwx /dev/sda3 /media/windows1
sudo mount -o rwx /dev/sda5 /media/windows2

Note that I am Using Ubuntu 11.10 and sda3 is my Windows C:, sda5 is G:.