How I can identify a device as a ROUTER or a MODEM?

Why would you want to, the devices are transparent so naming/identifying them is fairly pointless.

You could add an entry in /etc/hosts for the router. If your router's internal IP is 10.0.0.1 then the entry could be:

10.0.0.1 ROUTER router

You can now run:
ping router

and it will reply. Is this what you mean?

Please actionparsnip!!!!! Can you simplify your answer?

Please actionparsnip!!!!! Can you simplify your answer?

Please explain what you're trying accomplish and in which context. This may help giving you a better answer.

Why do you want them identifiable? The modem also is not identifiable as it is transparent and only serves as a decoder, the router's LAN IP can be used to identify it but as to why this is needed is at question. Why do you need this?

Oh, my friends are telling that ROUTES are better than MODEMS. That's why I'm trying to identify whether my device is a ROUTER or a MODEM.

Ok do you connect via wireless?
Does the device you connect to have multiple ports on the back?

No, I connect via wire and my device doesn't have multiple ports on the back.

Ok what make / model is the device?

My Router is Aztech and Model is ADSL2/2+ DSL705E. In the box of my router it is mention as "Aztech ADSL2/2+ ETHERNET MODEM ROUTER".

Then its a router, It just has an inbuilt modem. I have a cable modem which attaches to my router via a WAN port but yours is just an all in one solution.

Can you please tell me, what is the difference between MODEM and ROUTER?

http://www.differencebetween.net/object/difference-between-modem-and-router/

You could have found that easily.

My friends are telling that, ROUTER is better than MODEM. Is it true?

Depends what he means by 'better'. Without qualification it means nothing. What does he mean....

They tell me that, ROUTER is speeder than MODEM. Is it true?

No, a modem is actually insignificantly speedier than the router as there is no routing logic to be performed as there is only 1 way the data can go. The difference is massively insignificant (thousandths of second) and will not make any difference at all to the visible speed.

But, isn't that Modem is using USB port to connect to computer and Router is using RJ45 port to connect to computer?

You can connect to a router using USB if it supports it. USB 2.0 can reach 480Mbps whereas most home grade routers will use 100Mbps

This will not affect your web connection until your web connection exceeds 100Mbps.

You shouldn't get bogged down with how the PC connects to the router, but what features each device offers.

Can you PLEASE! tell me that, Does Modems too connect to computers via both USB and RJ45?

Yes. They can and do.
http://www.sbhacker.net/zencanada/images/categories/motorola_sb5100_.gif

Why are you so completely hung up on connectivity like this?

Because, I am new to internet and I want to know that, Do I use the best and speediest equipment to connect to the internet. And also in my day to day life I am thirsty to get knowledge about any field to it's deepest point. It is my way. PLEASE! Never mind about that. And I must say that, THANK YOU VERY MUCH!!!!! actionparsnip for helping me.

All home grade networking stuff will perform essentially the same speed, only the connection you pay for will affect the speed you see. Changing the router or modem will not make it any faster to any even remotely noticeable degree.