Do You Need Both a Wireless Router and Adapter for Xbox Live? Depends on Console Model

No. You need one or the other, depending on your console:

• Modern Xbox (360 S/E, One, Series): Built-in Wi-Fi → only a wireless router needed.
• Original Xbox 360 (no Wi-Fi): Requires both a wireless router and the Xbox 360 Wireless Network Adapter (or Ethernet cable).

The router creates the Wi-Fi network; the adapter (if needed) lets the console join it.