Kelly, 40, played for the Seadogs between 2013 and 2015, before starting his managerial career with Oldham Athletic.
He recently left Hyde United, having been moved upstairs in to a Sporting Director role over the summer.
The former Carlisle United, York City and Derry City defender will be in the dugout for Boro's next home league fixture against derby rivals Whitby Town on Saturday, and replaces John Deacey who left the club at the weekend.
Kelly said: "Firstly, I would like to thank the Scarborough Athletic board for this amazing opportunity, but obviously it is sad that a manager has lost his job.
"For me, all you have to do is look at the history of football in Scarborough to see how big a job this is.
"I played for Scarborough during their time at Bridlington and I got a taste of the passion that the fans have for the club.
"I love the challenge of being able to build something very successful.
"It will take time to implement how I work, but I believe that we can create something big.
"If we can get a few hundred back onto the gate and fans continue to come to watch, then in the next few years I don't see why we can't be pushing to be in the National League.
"I want the Scarborough to become a fortress. When I came here before as a manager I never really felt that.
"I want opposition players to be worried when they come to Scarborough because they'll know they are up against a well organised, fit team and a hostile crowd that backs the team 100%."
Chairman Trevor Bull added: "I'm absolutely delighted that we've been able to bring Darren to the football club.
"He is a young, ambitious manager that shares our belief that we can move this club forward in a positive way.
"Sometimes things all fall in to place and we decided that it was time for a change in manager. That coincided with Darren departing Hyde United.
"We made contact and arranged to meet him. We'd recently gone through a recruitment process where we had 50 applicants. At this stage of the season we needed to move quickly."