It's a micro-controller.
They only cost around $40 (compatible clones can be purchased for under $5)
It is both an educational tool and a serious controller for real-world applications
The development software is free to download and the device has a USB port making setting up for programming both cheap and easy.
It introduces students to programming concepts (but can support very sophisticated code as well).
On the hardware side it can connect to things like switches, temperature, movement, and infra red sensors. On the output side it can control things like LEDs, motors and door locks
The Arduino has a huge following with a very large online community support forum to help you with those tricky problems.
To get a better understanding, try searching on Youtube, there are thousands of tutorials and projects
Here's a simple project that controls a servo motor - only four wires to the Arduino needed!......
Some helpful links....
Beginner project - blinking LEDs