Pulse Width Modulation (PWM) is a technique for getting analog results with digital means. Digital control is used to create a square wave, a signal switched between ON and OFF. If you repeat this on/off pattern fast enough with a LED for example, the result is as if the signal is a steady voltage between 0 and VCC controlling the brightness of the LED.
Arduino is an open-source electronics platform based on easy-to-use hardware and software. Arduino boards are able to read inputs - light on a sensor, a finger on a button, or a Twitter message - and turn it into an output - activating a motor, turning on an LED, publishing something online. You can tell your board what to do by sending a set of instructions to the microcontroller on the board. To do so you use the Arduino programming language and the Arduino Software (IDE).