If you use a wireless modem, it will function as a router. Mine
works this way. The router is unneeded.
If you already have a modem but without wireless capability, you
need to get the router, and set up accordingly to the router's
manual.
And yes, an adapter is needed. Or a wireless LAN card.
In summary:
1. If already have a router, get an adapter or WLAN card for the
PC.
2. If have modem, but no wireless capability, get a router and the
adapter/WLAN card.