Running firefox browser from remote server but system always doesn't reflect the fact
 

I connected my remote server with1)
1st I open firefox browser on my local machine.
I do
and it outputs the firefox program which is fine.

Now, I open the browser on my remote server via terminal (running command). It takes more than 10 miniutes to come on my screen. And then the prompt comes back on the terminal so that I can run another command. Once browser comes, it works fine.
I run on remote machine but it doesn't
show the program.
(Later I concluded that this browser is running on my local box and not remote from where I ran)

2)
Now, what I do: I again disconnect from remote and reconnect.
And now rather than running the browser 1st from my local box, I ran from remote. This time (after 10 minutes) the browser appears on my screen but one can't work at all as it is too slow.
This time the prompt hasn't returned back. which returns if I run in my local box.
Now, running on my remote box (after opening a new terminal as former one hasn't returned prompt) outputs the firefox program. Note, earlier the command didn't give the output although I ran from remote.
After all this experiments, I concluded that at the 1st time, although I ran firefox from remote (which returned prompt), it was actually running on my local. And hence it was working fast. 2nd time it ran on my remote server and hence slow
(Just mentioning that my local internet speed is slow, while downloading I get a speed of 18 kb/s)

When you start Firefox it checks if it can find an already running instance of it. If it finds one it does not start another one, but tells the already running instance to open a new tab and display the wanted content there. Since running Firefox remote seems not to be something the developers thought about (honestly, it doesn't make much sense to do that) this has the side effect that this does not work correctly.
Regarding the low speed, with a connection of only 18kb/s any application (except those running the very old Motif toolkit) will run slow. Modern applications will render the content of their Window and then send it over the network to the display server. In that case using VNC or NX might be a better option, since they use compression, but even then your connection bandwidth will seriously limit you.

Thanks a lot for clarifying me. But one thing: rather than opening a new tab, it opens a new window.

Possible, but you will still have only one Firefox instance running.

Right TobiSGD, thank you.