I had this problem with my wife's iPad when I updated to 7.0.3 via OTA. It was easily fixed for me, and I assume it should work for everyone else.
It appears that sometimes during installation of the OTA update to 7.0.3, it gets stuck in recovery half way through.
You can connect it to iTunes and restore it, which will wipe it, OR you can download TinyUmbrella and kick it out of recovery. Once you do that, the installer should complete normally (it did for me).
1) Download and install TinyUmbrella for your PC/Mac.
2) Make sure iTunes isn't running and connect your device via USB.
3) Open up TinyUmbrella. Your device should show up on the left. Click on it and then choose "exit recovery".