It's both hardware and software, the various 802.11 levels are standards set for networking, it describes how the hardware and software work. 802.11b has a maximum speed of 11mbps, and 802.11g has a max of 54mbps. Either is most likely faster then your internet connection, so it won't speed up gaming.
If your router only supports 802.11b then you'll have to buy a new one to get 802.11g.
802.11b should work fine though.