There has been some buzz around ESP8266 Wi-Fi module, mostly because of its low price, and SDK availability, meaning it could become the Wi-Fi equivalent of ENC28J60 Ethernet module, and that for $5 you could potentially add Wi-Fi to your Arduino board for example. Since then, the price has come down even further, and if you are prepared to buy 5 pieces, you can now get the module for less than $3 / piece shipped, alternatively a single module costs $4, and a complete Wi-Fi + Arduino Uno (clone) kit goes for $15. A community has also been built around the chip, and a several project have been made with Arduino boards and ESP8266 module.
The best way to find information is to go to ESP8266 community forum, as well as read the Wiki on github. There’s currently a GCC toolchain for Espressif Systems ESP8266, open source tools for working with the firmware images and serial protocol, but the (leaked) SDK needs to be officially opened, as I understand it still requires an NDA.
ESP8266 does not have to be connected to another MCU board via its serial interface, and it can be used in standalone, as it also provides two GPIOs (version 2 only) so you can use it to control relays for example. The picture above is ESP-01, which is the most common module, but there are also other form factor for example with ESP-07 that’s even smaller but would require some soldering.
There’s been several project published on the web with Arduino + ESP8622, but AFAIK no libraries have been released yet, and people simply send AT commands in their sketches. You can check ESP8266 Wifi Temperature Logger project using Sparkfun Arduino Pro Mini 328, Seeeduino wrote a short tutorial with Seeeduino3 (Arduino UNO), and James Wolf did a short demo using ESP8622 and Arduino Micro board that fetch a URL, and display the HTML code and some of HTTP data in the serial monitor.
The sketch for the demo can be found here, and he also wrote some documentation.
Thanks to onebir for the tip.