RESOLUTION 1:
Check the PATH setting of the desktop PC where you are running Toad for Oracle and make sure you have the correct Oracle Home directory path specified at the beginning of the PATH system environment variable.
Here are the steps to get to your System Environment Variables:
1) Right-click Computer | “Properties” | “Advanced system settings” | “Advanced” tab.
2) Click “Environment Variables” button.
3) Under "System Variables" (lower part of the window) | locate PATH.
4) Highlight PATH | click on the Edit button | and make your changes.
5) Save and exit, and then re-test the issue.
RESOLUTION 2:
The error refers to 'oracore10.dll', which suggest your Oracle Client version is 10g. If you have a 64-bit Windows machine running 32-bit Toad and 32-bit Oracle Client, ensure you install Toad outside of the 'C:\Program Files (x86)\' folder (this is a bug in Oracle Client 10g). Install Toad in a different folder i.e. 'C:\Quest Software\'.