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. Odysseus can be used on some devices to downgrade to certain iOS versions providing the user has SHSH blobs and the device is jailbroken. OdysseusOTA can be used to downgrade an iPhone 4s or iPad 2 to 6.1.3 from any firmware providing the device is jailbroken even without SHSH blobs due to Apple not closing the signing of 6.1.3 OTA's for those devices. OdysseusOTA2 can be used to downgrade an iPhone5,2 to 8.4.1 the same way that OdysseusOTA did with iPhone 4s and iPad 2.
Reason for downgrading
Some users downgrade to older iOS versions since, for example, iOS 7.x does not run as well as older versions on outdated devices such as the iPhone 4, or iOS 9.x on the iPhone 4S. This could also be for nostalgia. There are many reasons as to why one would prefer an older version of iOS.
Restoring to Older Firmwares via Any Method
Downgrade iTunes to 11.0 or 10.7 and then restore when in PwnDFU.