I user to RedHat and Centos. I need to add storage to root partition, and I just don't understand what is happening.

My df -h:

guy@ubuntu:~$ df -h
Filesystem Size Used Avail Use% Mounted on
udev 2.0G 4.0K 2.0G 1% /dev
tmpfs 395M 424K 395M 1% /run
/dev/mapper/ubuntu--vg-root 23G 18G 4.1G 81% /
none 4.0K 0 4.0K 0% /sys/fs/cgroup
none 5.0M 0 5.0M 0% /run/lock
none 2.0G 0 2.0G 0% /run/shm
none 100M 0 100M 0% /run/user
/dev/sda1 236M 70M 154M 32% /boot

MY cfdisk:

 Disk Drive: /dev/sda
 Size: 26843545600 bytes, 26.8 GB
 Heads: 255 Sectors per Track: 63 Cylinders: 3263

Name Flags Part Type FS Type [Label] Size (MB)
----------------------------------------------------------------------------
 Primary Free Space 1.05 *
sda1 Boot Primary ext2 254.81 *
 Pri/Log Free Space 1.05 *
sda5 NC Logical LVM2_member 26585.60 *
 Pri/Log Free Space 1.05 *

and my fdisk -> p

Disk /dev/sda: 26.8 GB, 26843545600 bytes
255 heads, 63 sectors/track, 3263 cylinders, total 52428800 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 4096 bytes
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Disk identifier: 0x00092c8e

 Device Boot Start End Blocks Id System
/dev/sda1 * 2048 499711 248832 83 Linux
/dev/sda2 501758 52426751 25962497 5 Extended
Partition 2 does not start on physical sector boundary.
/dev/sda5 501760 52426751 25962496 8e Linux LVM

I know that there is no much free space now, buy yesterday I tried to extend to 60GB. There was space to extend, but I failed.

I need to extend the root partition:

/dev/mapper/ubuntu--vg-root

from cfdisk it looks like its /dev/sda5
But if fdisk it looks like sda5 and sda2 overlaps.

I don't have physical access to the vm - so GParted is out of the question.
What I don't understand it why there is an overlaps between sda2 and sda5?

When I tried to remove sda2 it removed sda5 as well. And the reboot failed.
When I removed only sda5 I couldn't start on the same block as it was in fdisk p.

What I want to know:
What is the difference between extendet and linux LVM volumes?
Why do they overlaps?
And is there a way for me to extend the root partition?

Why the overlap

In the early days of the PC, it was though that 4 partitions was enough for anyone.
Later we realised that we were wrong, so we added extended partitions, an extended partition contains another partition table, and some partitions. In Linux partitions 1→4 are the original partitions (named physical partitions, though they are no more physical than any other), and 5 onward are for the so called logical partitions, that sit inside one of the extended partitions (on of the physical partitions).

How to extend.

You may need to run from another OS (live), because editing a running file-system will not end well.

An alternative

Create a new partition with the free space.
Copy some directories to the new partition, and create symbolic links from the /root to the new partition.

E.g.

Mount new partition onto /big-partition, then

mv -T /home /big-partition/home
ln -s -T /big-partition/home /home

You can do this for other directories, as well. Use kdirstat, or other tool to find big directories, only move those that are not critical to OS operation.