Ai-Thinker ESP8266 WiFi Module ESP-01 1MB

This module offers you the possibility to connect your Arduino to WiFi for a very low price.

The ESP8266 module works with the normal UART port on your Arduino and can connect to networks, form your own network (hot spot) or both. The possibilities with this module are endless!

If you want to make a start, look here. In addition, you can find enough information at SeeedStudio, Instructables - ESP8266 Wifi Temperature Logger, ESP8266 forum, Hackaday and Nurdspace.
The module can also receive an update (more functionality, more stable, etc.). These sites are well described: ESP8266: Reflash Dance! en BikeAlive.

This variant has 1MB of memory instead of the 512kB on the standard ESP8266 module.