ESP8266 is a low-cost Wi-Fi microchip with full TCP/IP stack and microcontroller capability produced by manufacturer Espressif Systems in Shanghai, China (adapted from wikipedia...ESP8266).
WHAT IS NODEMCU?
NodeMCU is an open source IoT platform. It's based on the ESP8266 and is compatible with Arduino (adapted from wikipedia...NodeMCU). There are various versions of the NodeMCU and they are described here frightanic...boards.
GETTING STARTED WITH NODEMCU
1) Video tutorial for use the NodeMCU with Arduino IDE. For V1 version select the "NodeMCU 0.9 (ESP-12 Module)" board, otherwise "NodeMCU 1.0 (ESP-12E Module)" with V2 or V3.