To be able to connect over USB, there are several steps to follow.
First ensure that Android USB drivers are installed on your desktop computer. If you installed a software provided by your Manufacturer (like HTC Sync, Samsung Kies, Sony Ericsson PC Companion or Motorola Motoblur), the USB drivers are very likely installed on your computer.
When you open the Device Manager (open a Windows Explorer window, right click on the Properties of the computer icon, click on "Device Manager" on the left), when your device is plugged via USB, check that you have an entry with the word "Android". It can be for instance "Android USB devices" for HTC or "SAMSUNG Android Phones" for Samsung .
If you wish to install these USB drivers, you may have a look at the Android USB drivers section which describes how to download and install USB drivers for Android devices for each manufacturer.
When you connected your Android device the first time, Windows may have automatically installed a driver. It doesn't mean that Windows installed the correct driver. Actually you may have a driver that allows you to browse your files on your Android device but doesn't allow BirdieSync to connect via USB.
So, as a major advice, it is strongly suggested to manually (re)install Android USB drivers downloaded from your Manufacturer web site.
On your Android device, on your Android home screen, if your Android version is 2.x - 3.x, open menu "Settings/Application/Development" and check "USB debugging". If your Android version is 4.x or later, open menu "Settings/Developer options" (must be switched ON with 4.1 and later) and then check "USB debugging". This option is not harmful and all applications which need to connect over USB need this option to be checked.
Check that your PC firewall authorizes both incoming connections for adb.exe and BirdieSync.exe processes.
To ensure that your firewall doesn't prevent from connecting your Android device to the PC, temporarily disable it and wait for 1 mn before trying to connect again.
If you succeed in connecting once your firewall is disabled, then check again that adb.exe and BirdieSync.exe are properly authorized. If you use Windows firewall, you may try to reinstall BirdieSync so that Windows firewall rules are declared again to authorize BirdieSync to accept incoming connections. If you use a Public firewall profile and your PC is on a local network behind a router, try switching to a Private firewall profile which is less restrictive.
If you use a firewall on your Android device, you may also try to temporarily disable it to ensure it doesn't prevent your Android device to connect. Your Android device must authorize BirdieSync app and adbd daemon.
If your mobile device has a recent version of Android, it may require the latest version of adb.exe to be able to connect. So install the latest version of BirdieSync to ensure you use a recent version of adb.exe.
Android 4.2 devices require at least adb.exe 1.0.31 which has been provided since BirdieSync 18.104.22.168 supporting Thunderbird 17.0.
To test that Google adb.exe process can detect your mobile device via USB:
Open Command Prompt Windows (Windows Start menu/All programs/Accessories/Command Prompt) and type:
If you have Windows 64 bits:
cd C:\Program Files (x86)\BirdieSync\Android\Adb\1.0.31
If you have Windows 32 bits:
cd C:\Program Files\BirdieSync\Android\Adb\1.0.31
You should then see your device listed. BirdieSync is not involved in this detection process. Only Google adb.exe is. If your device is not detected, there is very likely a problem with your Android USB driver or USB debugging is not enabled on your Android device.