GRUB in MBR or Partition?

where should i install GRUB? MBR or Partition Boot Sector?

What are the advantages and disadvantages of each approach?

Also, if we install the GRUB in the Partition Boot Sector then how the boot process works?

Read through the first half of this answer to understand the basics of MBRs, VBRs (aka partition boot sectors) and boot managers.

Now that you've read that, you understand that you need some boot loader in the MBR, or the BIOS can't boot from the drive. The BIOS loads code from the MBR, which then loads code from a VBR (or sometimes provides a menu of VBRs to choose from).

If you install Grub to a VBR but not the MBR, you'll need something else in the MBR capable of chainloading the Grub bootloader. If you have something else in mind (for example, Win7's boot manager, or another version of Grub, or some other boot manager), it's certainly possible to do, but specifics will depend on the other boot manager's capabilities.

Any analysis of advantages vs. disadvantages will have to compare Grub's features to those of other specific bootloaders. You will need to be more specific as to what your options might be.

In the old days of LILO and GRUB Legacy (aka just GRUB), it was usually considered a matter of preference where to install your Linux bootloader. MBR or partition would work about the same, assuming for the partition case (as per quack quixote's answer) that the standard DOS MBR would chainload the first partition marked "bootable" in the partition table. (I assume current Windows MBRs still do this, though I'm not sure.) Installing to a partition was considered slightly safer as it left the original boot path untouched.

But these days you are likely using GRUB2, which has a completely new design with many more capabilities and greater robustness. GRUB2, however, advises against installing to a partition, as some of the reliability features are lost. Generally, this configuration is no longer encouraged, and may not be supported by your operating system. I'm not enough of a GRUB expert to understand the full implications, but unless you feel like taking the risk and supporting yourself, I would install GRUB2 on the MBR. It is very reliable and will almost certainly boot your other operating systems.

Update: It looks like the default MBR on today's computers may not be as simple as it used to be. After installing GRUB on the MBR of my ThinkPad X220, I found that pressing the "ThinkVantage" button during boot no longer got me to rescue mode. That feature was in the original MBR. So we're in a bad spot: GRUB and the Linux distributions recommend installing to the MBR, ignoring the fact that the existing MBR may have features the user is relying on.