Sheridan, 51, led Oldham to 17th in League One after taking over from David Dunn in January.
Ex-Nottingham Forest goalkeeper Mark Crossley will be assistant to Sheridan, whom he worked with at Chesterfield.
Previous manager Mark Cooper left Notts in May to join Forest Green, despite the offer of a permanent contract if a points target was achieved.
Oldham initially rejected approaches from Notts County for Sheridan but later allowed him to speak to Notts, who finished 17th in League Two and are currently under a temporary transfer embargo because of unpaid bills.
"There are several reasons behind this move and not all will be apparent to everyone, but John goes with my blessing and eternal thanks for his tremendous effort," Latics chairman Simon Corney told his club's website.
"Whilst we recognise the huge disappointment many fans will be feeling, we ask everyone to try to stick together and continue to support our club. This is not something we wanted or envisaged."
Sheridan started his managerial career at Oldham in 2006 and left after three years with the Latics.
While in charge of Chesterfield, he won the League Two title in 2011 and the Football League Trophy a year later.
"Notts County is a big club - far bigger than its current league position suggests and I know what is required to succeed and get the club back to a higher level," Sheridan told Notts County's website.
"The tremendous home support has always made Notts a tough place to come - this passion can make a real difference and give us a huge advantage in our endeavours next season."
Chief executive Jason Turner added: "I'm confident that we've found a manager who can have us challenging for promotion at the top end of the League Two table.
"He's vastly experienced with a proven track record at this level, so to attract somebody of his calibre is a real statement of intent by the club."
John Sheridan has left Oldham Athletic to become Notts County's new manager on a three-year contract.