Apple doesn't sign the firmware anymore. If your device has an S5L8920 or S5L8922 SoC or an A4 to A11 SoC and you have saved your SHSH blobs for the firmware you are restoring to, you are doing something incorrectly. If you don't have SHSH blobs for the firmware you are restoring to, there is nothing you can do. If you are restoring to a device with an A12 or newer SoC, there is no (publicly disclosed) bootrom exploit to restore to older versions of iOS. You should still save your SHSH blobs in case an exploit is discovered. However, Odysseus, OdysseusOTA or OdysseusOTA2 can be used if the user has SHSH blobs, but it only works on some iOS versions and requires the device to be currently jailbroken.
The (Apple TV/iPhone/iPod/iPad) "XXXXXX" could not be restored because the firmware file is not compatible.[edit source]
You are using the wrong firmware file (or it's corrupt). Download the correct firmware file that matches your device. The firmware file's filename will contain the internal identifier of the device it's intended for.
To determine which firmware should be downloaded, find your device's model numbers on Models. Look at the "Identifier" column; when downloading firmwares, the URL must contain that string. Alternatively, if you have iTunes 11 or later, you can click the device and click where the serial number is three times to show the "Product Identifier", which will show you what firmware you need. Once you have the identifier, visit Firmware and find the version you want for your device.
If you want to upgrade or downgrade the baseband, make sure you didn't change the hosts file to add the Cydia Server. Cydia cannot handle the requests for the baseband certificates. Maybe you tried to use SAM Activation method.
There was a problem downloading the software for the (Apple TV/iPhone/iPod/iPad) "XXXXXXX" the network connection was reset, make sure your network settings are correct and your network connection is active, or try again later.[edit source]
You need to disable both your Anti-Virus and your windows/security firewall in order for the update/restore to go through.