Which thermodynamic law does the article talk about breaking?  Explain how this would break that law.

The article talks about breaking the second law of thermodynamics. 

Does it really break that thermodynamic law?  Explain.

No, it didn't. The temperature did drop, but later it slowly increased in temperature. 

What applications could this technology be used for in the future?

This technology can be used to minimize electric consumption, therefore using less energy to lower temperature. 

What applications is this technology used for currently? (You will need to search for this answer outside the article.)

This technology is currently used in mini-fridges that can be placed in cars/vans, or hotel rooms.