Firmware downgrading

Downgrading is the process of restoring to an older firmware. Apple does not allow downgrading unless the firmware is being signed. Previously, downgrading was only possible on older devices that have SHSHs saved and are vulnerable to bootrom exploits such as the limera1n Exploit. However, it has become possible to downgrade devices with no (known) bootrom exploits to certain firmware versions, such as remaining on iOS 5.x or downgrading to iOS 6.1.3. It is also possible to downgrade an iPad 2 to 5.x with 4.x and 5.x SHSH or 4.x with only 4.x SHSH blobs. iPhone 4s and iPad 2 downgrading will require redsn0w.

Reason for downgrading
Some users downgrade since iOS 7 does not run as people want on older devices like the iPhone 4. This can also be for nostalgia. There's many reasons to why one would want an older iOS.

Restoring to Older Firmwares via Any Method
Downgrade iTunes to 11.0 or 10.7 and then restore when in PwnDFU.