In short:
If you have a 32 bit Toad, you need a 32 bit Oracle client.
If you have a 64 bit Toad, you need a 64 bit Oracle client.
It's usually OK to have multiple Oracle clients installed, but occasionally they clash. So if you have problems, you may want to install all of them, and then reinstall only what you need. The following link has tips on that process.
In short:
If you have a 32 bit Toad, you need a 32 bit Oracle client.
If you have a 64 bit Toad, you need a 64 bit Oracle client.
It's usually OK to have multiple Oracle clients installed, but occasionally they clash. So if you have problems, you may want to install all of them, and then reinstall only what you need
There are some tips on how to do that here: