Update :- My laptop doesn't support 64 bit OS and it's confirmed. I have a desktop though not so powerful as my laptop which has E2200 Intel processor and my motherboard is 82801G Intel. Can anybody now tell me whether this motherboard and processor supports 64 bit OS or not?
You could just Google E2200 Intel and see if you can get a link to its specs.
Intel might also have a (free) utility you could download which would list the capabilities of your CPU. (I didn't look).
If it was me, I would check using CPU-Z, a slightly more technical free utility that I and lots and lots of others are fond of. It's a very bare bones Windows program that doesn't really require an "install". You just run the program.
The image above is a screen shot of CPU-Z from its web site. What you want to look for is EM64T in the "Instructions" box. This indicates support for the x86_64 64-bit Instruction Set Architecture (ISA). This is what is required to install a 64-bit version of Windows (or Linux for that matter). If it's not there then you can only run a 32-bit OS.
I forgot that I was going to mention that 64-bit versus 32-bit is a subject that people bring up quite often on this site.
You should see a list of links to questions similar to yours along the right hand side of this page. If you didn't get enough info in the answers here or even if you're just curious then I encourage you to browse through the other/earlier questions along this line.