Programming esp8266 NodeMCU (with Arduino IDE)
Any ESP8266 board can be programmed in the Arduino environment, provided you install the ESP8266 addon.

See here for instructions:
https://learn.sparkfun.com/tutorials/esp8266-thing-hookup-guide/installing-the-esp8266-arduino-addon

The NodeMCU is just a name of one of the boards that are populated with the ESP8266 MCU. You have other names such as Wemos which is another popular board.

The programming language in Arduino IDE is a language that is merely a set of C/C++ functions, which are then compiled to a binary file which is uploaded to the Microcontroller's Flash memory.

IF you want to use LUA, there are a few IDEs that can be used such as ESPlorer (https://esp8266.ru/esplorer/), but other are available as well.

You can find more info on the ESP8266 here:
https://www.esp8266.com/wiki/doku.php