VirtualBox does not start on Mac OS X

All your VirtualBox VMs and the application's settings are not stored in the actual .app file. You can try reinstalling it by just running the installer again. This is also how you would update VirtualBox to a newer version.

Your original VMs and settings are stored in ~/VirtualBox VMs, ~/Library/VirtualBox and ~/Library/Preferences, respectively. You can of course make a backup of those if they are really important to you, and if you haven't yet it's really a good time to set up Time Machine.

To resolve this on El Capitan when using Virtualbox versions lower than 6.x run the following from terminal:

for bin in VirtualBox VirtualBoxVM VBoxNetAdpCtl VBoxNetDHCP VBoxNetNAT VBoxHeadless; do
    sudo chmod u+s "/Applications/VirtualBox.app/Contents/MacOS/${bin}"
done