The former Magpies captain — who played 128 times for the club — spent last season in charge of Hungarian side Ferencvaros.
Short replaces Steve Cotterill who left Meadow Lane last month at the end of his short-term deal after guiding the club to the League Two title.
The 41-year-old said: "I'm a little bit shocked to be honest, having seen some of the names that have been linked with the job.
"I actually rang [club life president] Derek Pavis a couple of years ago enquiring about a job here and he gave me the best piece of advice — to go and get some experience.
"I took that on board and, whilst I was reluctant to go abroad to gain that experience and it would prove to be the toughest two years of my life, it has paid off because it has brought me to where I am today.
"I am delighted that the chairman is prepared to give a hungry young manager a chance and I feel that I have shown the desire to succeed here.
"If I can enjoy half as much success here as a manager as I did as a player, it's going to be a fantastic time for myself and the club."