ESP8266-01
Simple, low cost UART WiFi chip. Espressif Firmware supported.
Hello World
Import programmbed-os-example-mbed5-wifi
A quick example of a simple WiFi application using the WiFi and network-socket APIs that is provided as a part of mbed-os.
Library
Import libraryesp8266-driver
The driver for the ESP8266 WiFi module
Datasheet
http://www.seeedstudio.com/wiki/Grove_-_UART_WiFiNotes
The ESP8266 is a convenient and cheap UART to WiFi module. Due to its low pricepoint (<5$) and ease of use (any board with UART) it is a popular choice for easily adding WiFi support to any project.
Change_Firmware
The ESP8266 Breakout board supports multiple firmware's. Currently the two major firmware vendors are Espressif and AT Thinker. These different firmware's are incompatible. For instructions on how to update your board to use a selected firmware see the Firmware Update page.
Espressif
The Espressif firmware is from the company who makes the ESP8266, as such it has the best support and fully supports both TCP and UDP. The mbed team uses this firmware as it is the most stable. The libraries supporting this firmware comply with the Network Socket API and are compatible with the future upgrade path for mbed and its socket model.
Other cool projects
These are some community contributed projects that use the ESP8266 in awesome ways
Wiznet Dual Network Wifi
A dual network interface project(WIZnet TCP/IP and ESP8266)
TLDR;
Upgrade your ESP8266 board to use the Espressif firmware and use the Espressif examples. It's more awesome and future proof.