Explain what is meant by "Arduino core"
The Core is the software API for a specific group of chips. 

For instance, the AVR core supports the AVR chips found on most Arduino boards. The SAM3X core supports the Arduino Due. Etc.

It is the code that provides things like digitalRead(), analogWrite(), millis() etc. 

Since each family of chips works differently that abstraction layer between your sketch and the physical hardware of the chip needs to be different.

In your default IDE installation the Arduino AVR core can be found within the hardware/arduino/avr/cores/arduino folder.

Cores are usually installed as part of a "boards" bundle in the boards manager. This includes definitions for the boards ("variants"), the core and, equally importantly, the compiler and firmware uploading software for the chip on the board.