I synced with iPhone and was asked whether I wanted to upgrade my iPhone 5. After that I could not open my phone. When pressing the started button I get a picture of what looks like a charger with an arrow pointing to iTunes. I tried to reboot, but to no avail.
When you see the lightning cable and iTunes logo on your screen, it means that your iPhone has entered Recovery mode.
The best thing to do is to connect your iPhone to iTunes on a Mac or Windows PC and restore it. ITunes should display a popup saying that you connected an iPhone in Recovery Mode.
From the screen in iTunes, select restore or restore and update. (Actual button names could be different as what I say)
After you restored your iPhone you need to configure it. If you want, you can restore a backup from iCloud or iTunes when there's one available.