partitioning

1. First you need some unallocated space to create the partitions for each mountpoint (/var, /home, /tmp). Use Gparted for this.

2. Then you need to create the filesystems for those partitions (can be done with Gparted too) or use:

mkfs.ext4 /dev/sdaX
for example to create a new ext4 filesystem on the /dev/sdaX device (replace /dev/sdaX with your own device)

3. Mount the new filesystem under /mnt

mkdir /mnt/var        
mount /dev/sdaX /mnt/var
4. Go to single-user mode so that there is no rw activity on the directory during the process

init 1
5. Enter your root password.

6. Backup data in var only (not the /var directory itself)

cd /var
cp -ax * /mnt/var
7. Rename the /var directory after your data has been transferred successfully.

cd /
mv var var.old
8. Make the new var directory

mkdir var
9. Unmount the new partition.

umount /dev/sdaX
10. Remount it as /var

mount /dev/sdaX /var
11. Edit /etc/fstab file to include the new partition, with /var being the mount point, so that it will be automatically mounted at boot.

/dev/sdaX       /var     ext4    defaults    0 0