There's 3 things that need to happen to boot up the ASIAir.
The Raspberry Pi has to find and start the kernel, this is on the boot partition of the sdcard which is formatted VFAT. You should be able to see this partition on a Windows machine. The boot partition gets mounted read-write because it's where the images are written (if you don't use a USB) and can get corrupted if you power off without first shutting down from the app.
The root partition is then mounted, this is the second partition on the sd card and is a Linux format filesystem, it is mounted read-only so doesn't tend to get corrupted so easily. This is where the Raspbian OS is installed.
The last thing that happens is rc.local starts the asiair.sh script, this is on the third partition which is also a Linux format filesystem. This partition is mounted as /home/pi and is where the ZWO proprietary software lives. This script sets up the network and starts the imager and guider daemons.
The easiest way to diagnose problems with the sdcard not booting is to connect it to a Linux machine. If you don't have one, download a live USB stick of Linux Mint (Cinnamon), you can then just plug it in and reboot to Linux, it doesn't mess with your Windows disks or installation. When you connect the sdcard the partitions should automatically mount, if they don't, then the sdcard might be corrupted. Look for errors by typing dmesg. You can fix lightly corrupted filesystems with fsck but if files are missing, you need to reinstall them.
You can also mount the partitions from the ASIAir image file with losetup.
