Wireless router usually connects to a modem, the modem gets internet from Internet Provider, but normally only one device can connect to the modem and get internet. So the router connects to the modem, and it can share the internet to multiple devices. Some Wi-Fi routers can provide Wi-Fi networks, then phones and laptops can connect to it. But some desktop doesn't have Wi-Fi function, cannot connect to the router wirelessly, so we can plug a wireless adapter into the desktop, to make the desktop has Wi-Fi capacity.

I will try to answer your question. First of all, the difference between a Wi-fi adapter and a Wi-fi router in the tasks they perform. The adapter is a discrete receiver, with the help of which the computer can connect to the Wi-Fi network. Router this is a device, which will organize this network. Router and adapter сannnot connect individually to the network, they work only together.