Ai-Thinker ESP8266 WiFi Module ESP-12F 4MB

This module offers you the possibility to connect your Arduino to Wi-Fi 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 (in Dutch), 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 stablity, etc.). These sites are well described: ESP8266: Reflash Dance! en BikeAlive.

This variant has 4MB memory and must be soldered. The pins on the sides are usable for general use, the pins on the bottom (6 pins on the bottom of the second picture) can partly be used to program the flash chip. (these do not offer extra GPIO pins)

This datasheet contains more specific information if needed. (ESP-12E broadly corresponds to the renewed ESP-12F)

This ESP-12F is an improved model with respect to the ESP-12E. The module has an improved antenna and some adjustments have been made to the PCB for better Wi-Fi performance.

Note: This module is delivered without a pre-programmed firmware!