The pioneering businessman, who was the mind behind the revolutionary iPhone and iPad devices, had been battling pancreatic cancer.
He stepped down from his post as Apple's chief executive in August saying he could no longer handle the job due to his illness.
Mr Jobs, who lived in Silicone Valley, California, died on Wednesday surrounded by his family..
Apple said it was "deeply saddened" by the news.
Prime Minister David Cameron and Labour leader Ed Miliband both paid tribute to Mr Jobs using Twitter.
Mr Cameron said: "Steve Jobs transformed the way we work and play; a creative genius who will be sorely missed. Our thoughts are with his family."
Mr Miliband said: "Very sad to hear about the death of Steve Jobs. His work had a huge impact on our lives and he inspired thousands of young minds."
A company spokesman said: "Steve's brilliance, passion and energy were the source of countless innovations that enrich and improve all of our lives. The world is immeasurably better because of Steve.
"His greatest love was for his wife, Laurene, and his family. Our hearts go out to them and to all who were touched by his extraordinary gifts."
Bill Gates, the founder of rival technology company Microsoft and friend of Mr Jobs, said he would miss him "immensely".
"I’m truly saddened to learn of Steve Jobs’ death. Melinda and I extend our sincere condolences to his family and friends, and to everyone Steve has touched through his work.
"Steve and I first met nearly 30 years ago, and have been colleagues, competitors and friends over the course of more than half our lives.
"The world rarely sees someone who has had the profound impact Steve has had, the effects of which will be felt for many generations to come."
Mr Gates added: "For those of us lucky enough to get to work with him, it's been an insanely great honour. I will miss Steve immensely."
US President Barack Obama also paid tribute: "Michelle and I are saddened to learn of the passing of Steve Jobs. Steve was among the greatest of American innovators - brave enough to think differently, bold enough to believe he could change the world, and talented enough to do it."
Facebook founder Mark Zuckerberg wrote on his Facebook page: "Steve, thank you for being a mentor and a friend. Thanks for showing that what you build can change the world. I will miss you."
Taoiseach Enda Kenny said his legacy will be about the visionary products created and how he changed they way people think and feel about technology.
"Steve Jobs was a creative genius who broke down walls in business and opened doors in people's minds," Mr Kenny said.
"His innovative prowess in the area of technology has brought about a level of access to information for millions that few would have ever foreseen.
"His legacy will be not just his products and business achievements, but also the way in which he altered mindsets in the business world and in everyday life."
Apple has a large campus in Cork which opened in 1980 to manufacture the Mac computer. It employs about 1,400 people and has expanded into all aspects of the company's business.
In his own words: 'Find what you love'
No one wants to die. Even people who want to go to heaven don't want to die to get there. And yet death is the destination we all share. No one has ever escaped it. And that is as it should be, because Death is very likely the single best invention of Life. It is Life's change agent.
It clears out the old to make way for the new. Right now the new is you, but someday not too long from now, you will gradually become the old and be cleared away. Sorry to be so dramatic, but it is quite true.
Your time is limited, so don't waste it living someone else's life. Don't be trapped by dogma — which is living with the results of other people's thinking. Don't let the noise of others' opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition.
They somehow already know what you truly want to become. Everything else is secondary.
Mr Jobs helped to set up the company in 1976 after dropping out of college.
In October 2003 he was diagnosed with pancreatic cancer, although details emerged only the following year after he revealed he was to step back from the company to fight the disease after a special diet had failed to cure it.
Fears about his health resurfaced several years later, with commentators noting his gaunt features and weight loss.
In 2009, days after denying his ill health to shareholders, Mr Jobs revealed he was to take a leave of absence of six months. His exact issues were not disclosed, although it emerged his treatment had culminated in a liver transplant.
He returned to the company that summer and made his first public appearance in September of that year, as ebullient as ever about the company's products.
During each of Jobs's medical absences, Cook had been handed the Apple reins and is seen by company insiders, shareholders and analysts as a safe pair of hands. "I have great confidence that Tim and the rest of the management team will do a terrific job executing the exciting plans we have in place for 2011," Jobs said.
While Cook was in charge, Schiller was the public face of Apple. A charismatic character who joined Apple in 1997, he took over Jobs' highly lauded keynote presentations at product launches during previous absences.
From Macintosh to the iPad
1976 Jobs co-founds Apple
1980 Apple's stock market flotation is biggest since Ford in 1956
1985 Leaves Apple to concentrate on new ventures, including Pixar and NeXT
1997 Returns to Apple promising to revitalise the flagging company
2000 Resumes as Apple CEO
2002 iPod launched
2003 iTunes launched
August 2004 announces he has been diagnosed with pancreatic cancer and has undergone surgery to remove tumour.
2007 iPhone launched
June 2009 Returns to work six months after taking time off for liver transplant as part of cancer treatment
January 2010 iPad launched
17 January 2011 Announces he is taking a break from day-to-day operations to concentrate on his health.
11 March 2011 iPad 2 launched
24 August 2011 Jobs resigns as Apple CEO
05 October 2011 iPhone 4S launched
06 October 2011 Apple announces the death of Steve Jobs