If you have enabled AP Isolation/Client Isolation on your Wi-Fi router, or if this is the default setting, you will need to turn it off. Learn more about AP Isolation here. If the network you want to connect to is hidden (a network that is set not to broadcast its name), you'll need to manually input the network's name and password. Please call your network provider if you don't know the name or password of your Wi-Fi router.
If you wish to connect your Chromecast without using Wi-Fi and have an available Ethernet connection nearby, you might benefit from using the Ethernet adapter.