Ai-Thinker ESP8266 WiFi Module ESP-01S 1MB

This module offers you the possibility to connect your Arduino to Wi-Fi for a very low price. This "S" version of the ESP-01 module has minor improvements in the PCB compared to the normal ESP-01 module.

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 Wi-Fi Temperature Logger, ESP8266 forum, Hackaday and Nurdspace.
The module can also receive an update (more functionality, more stable, etc.). On these sites that process is well described: ESP8266: Reflash Dance! and BikeAlive.

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