My recollection on the subject:
1) Suggested/documented by Intel: use the Intel Galileo MiniPCI Express slot and an Intel® Centrino® Wireless-N 135, as explained at How to Set Up a Wireless Card for Intel® Galileo Boards.
2) A more expensive, but almost "zero software & always running" combination of cable and wireless router"
Intel Galileo Ethernet <=================> wifi router <~~~~~~~~~~~~~~~~~~~> Internet & wifi accessible "things"
Ethernet cable wifi communication
3) Using a module for RF communication, most of the examples I know of are using the nrf24l01+ (nRF24L01 / 2.4GHz RF / Products / Home - Ultra Low Power Wireless Solutions from NORDIC SEMICONDUCTOR), but several other options are available. I've used myself in some simple projects such as in Intel Galileo - Arduino Car - YouTube
4) Using a module for wifi communication, most of the examples I know of are using the ESP8266, such as briefly explained at Wi-Fi module ESP8266 in Galileo
5) Using a module for bluetooth communication, which would be not much more than replacing a ESP8266 by a bluetooth module, as in video3 - YouTube (approx. at sec. 15 appears a well known bluetooth module used in many Arduino examples, and the complete Intel Galileo "system" is controlled from a tablet, connected to the Galileo via bluetooth).
If there is some other way of wireless comm. with the Intel Galileo please let me know.