To access remote machine as if using local machine.
 

I recently purchased a server of mine (VPS).
I log in via ssh and so enter into the terminal of my
remote server which is Centos 5
Now please assist me to say how I can access it as
if I am assessing my local machine.
By this I mean, I should see the complete desktop.
Then i can open terminal, browser,etc. just like my personal computer here locally.
(Just for info, mentioning that my local machine OS is ubuntu13.04)

I would think you could use tigervnc but configure it to use tunneling over ssh.

I googled and found that tightVNC mentions that it is good for low speed internet and tigerVNC mentions that it is focused on performance and remote display functionality.
My internet speed is too slow (18kbps while downloading). So, which one is better tightVNC or tigerVNC?

Since you know which one is better over a slow link, which is better over a faster link, and know your connection speed, should you be able to figure this out for yourself??.

Slow link = tightVNC, as documentation says. Also, since you've said you have years of experience in unix, you should also already know that VNC and remote-desktop apps are insecure, slow, and unnecessary, and that ALL of your administration can be done over SSH. GUI programs can be run using SSH forward.

question #1 to ask, the VPS you are renting does it have the GUI installed? If not then you will not see anything remotely like you see in Ubuntu. its that simple.

as you should have root access you can yum install gdm and gnome shell (do some google searching as ive never bothered with a GUI for my centos servers)

Once you have it up and running you can either, as mentioned above, run some form of VNC or rdesktop, as that is built into the GUI in most cases, or just use ssh -Y or -X to access the specific local application. ie: firefox or thunderbird, or gedit, etc...

with that slow of a connection, neither will work well. you will be better off running raw ssh -X or ssh -Y on the single application you need access to. see my post above and tbones.

I doubt you'll find anything that will give you a useable remote GUI desktop with that slow of a connection. I have never found VNC (any flavor of VNC) to be fast. NX is significantly faster, and more secure. But still, that connection speed is not good for this type of stuff.

Yes you pointed out the right thing.
It seems the below error is because GUI isn't installed.
I installed gparted and then ran only to get the below message.

You got it, because you haven't set up your X environment yet. Going back to what I posted before about using X over SSH, you have to set up the DISPLAY variable, and also configure ssh to ALLOW X to run over it, as well as using the "-x" parameter when you make your connection. Did you do any of that?

Thanks TBOne for the method to resolve the issue.
No, I haven't done any of that.
I need to google and understand how to proceed.
Any further assistance in this regard w'd be appreciated.

Pleased to read this.

I w'd like to update that I have changed my Operating System from Centos 5 to Ubuntu13.04 in server (VPS).So, I have the same OS locally and in VPS.

then you need to talk with your VPS provider. this is something they should handle for you as they will have the tools to manage the VM that you will not.