He had been out of work since parting company with the Robins in November.
Crawley, who were relegated from League One earlier this month, had been without a manager since John Gregory left the West Sussex club on 9 May.
"My job is to put a competitive squad together that will challenge at the right end of the table," 45-year-old Yates told the Crawley website.
He spent almost five years in charge of Cheltenham and guided them to the League Two play-offs in 2012 and 2013.
After successive top-seven finishes and unsuccessful play-off campaigns, Cheltenham finished 17th in League Two in 2013-14.
And ex-Kidderminster boss Yates was then sacked late last year, following a run of four successive league defeats that left them 18th.
He was the third-longest serving manager in England's top four divisions at the time of his departure from Whaddon Road.
"I have been out of the game for a few months and when this opportunity came up it really excited me," he said.
"There's a big challenge ahead but one I am really looking forward to."
Gregory, 60, parted company with Crawley earlier this month, having spent the past six months recuperating from heart surgery.
Dean Saunders was placed in interim charge of the Reds in December, but was unable to save the club from the drop to League Two.
The 50-year-old former Liverpool and Aston Villa striker was appointed Chesterfield manager last week.
Crawley chief executive Michael Dunford said that Yates stood out among "some very strong candidates".
He added: "He has a very deep knowledge of the level we find ourselves at next season and is a first-rate coach.
"The board spoke to several people in the game who have worked with Mark and they were all very positive about his credentials, his knowledge and his enthusiasm."

The charge relates to a $44m (£29m) payment to German banker Gerhard Gribkowsky of Bayern Landesbank, linked to the sale of a stake in F1.
Mr Ecclestone denies bribing Gribkowsky, and says the money was intended to stop the banker from exposing him to a UK tax inquiry.
Mr Gribkowsky was sentenced to eight-and-a-half years in jail in Munich.
Speaking to the Financial Times on Wednesday, Mr Ecclestone said: "I have just spoken to my lawyers and they have received an indictment. It's being translated into English."
"We are defending it properly. It will be an interesting case. It's a pity it's happened."
Mr Ecclestone has six weeks to respond to the indictment, Formula One Group said in a statement.
The charges are related to the 2006 sale of a 47% stake in Formula 1 by Bayern Landesbank, which had acquired it from the bankrupt Kirch media group.
Gribkowsky had been in charge of managing the sale to private equity group CVC Capital Partners.
Bernie Ecclestone - the man, the myths and the motors
CVC has since reduced its stake in a series of deals. It has not commented on the indictment.
At his trial last year, Gribkowsky admitted corruptly receiving $41.4m (£26.6m) in bank commissions and a large payment via the Ecclestone family's Bambino Trust.
He maintained he had been paid to undervalue the shares.
But Mr Ecclestone says he paid Gribkowsky to keep him quiet, believing the banker was planning to give false information to the UK authorities about his tax affairs.
Mr Ecclestone said at the time that proving the allegation was false "would have been very expensive for me".
"The tax risk would have exceeded £2bn. I paid him to keep calm and not to do silly things," he said.
Mr Ecclestone has until mid-August to respond to the charges. The court will then decide whether to take the case to trial - it could be heard by mid-September.
He faces a possible jail term if found guilty, something he has previously said would force him to stand down as head of F1.
His lawyer told the Financial Times the indictment was largely based on statements from Gribkowsky, and that Mr Ecclestone intended to dispute these.
Crawley Town have appointed former Cheltenham boss Mark Yates as their new manager on a two-year deal.
[NEXT_CONCEPT]
German prosecutors have indicted Formula 1 motor racing boss Bernie Ecclestone on a bribery charge.