A Wireless G Adapter refers to a wireless network adapter that supports the 802.11g Wi-Fi standard. The "G" stands for 802.11g, which is a Wi-Fi standard that was widely used for wireless networking before the newer standards (like 802.11n, 802.11ac, and 802.11ax). The Wireless G Adapter allows your computer to connect to Wi-Fi networks using the 802.11g protocol, providing speeds up to 54 Mbps (megabits per second), depending on the quality of the connection.
A Windows driver for a Wireless G Adapter is software that allows the operating system (Windows) to communicate with and control the hardware (the Wireless G Adapter). Without the correct driver, the adapter won't work properly, or at all, in Windows.