The former Nasa astronaut had suffered complications from heart surgery he underwent earlier this month, his family said.
He famously uttered the quote moments after setting foot on the lunar surface: "That's one small step for man, one giant leap for mankind."
Astronomer Sir Patrick Moore said: "As the first man on the Moon, he broke all records. I knew him well. He was a man who had all the courage in the world."
Physicist Professor Brian Cox tweeted: "Sad to hear about death of Neil Armstrong. I do think Apollo was the greatest of human achievements. For once, we reached beyond our grasp."
Armstrong commanded the Apollo 11 spacecraft that landed on the Moon on July 20 1969. He and fellow astronaut Edwin "Buzz" Aldrin spent nearly three hours walking on the Moon, collecting samples, conducting experiments and taking photographs.
An estimated 600 million people - a fifth of the world's population - watched and listened to the first moon landing, the largest audience for any single event in history.
Armstrong and then wife Janet would later meet Queen Elizabeth and Prince Andrew at Buckingham Palace at a reception following the moon-landing in 1969 during his 22-nation 38-day world tour.
It continued: "For those who may ask what they can do to honour Neil, we have a simple request. Honour his example of service, accomplishment and modesty, and the next time you walk outside on a clear night and see the Moon smiling down at you, think of Neil Armstrong and give him a wink."
Virgin boss, space enthusiast and Oxfordshire resident Sir Richard Branson said Armstrong was an inspiration for generations.
"He was an extraordinary individual who had achieved an absolutely extraordinary thing in his lifetime."
"He was quite a shy, very quiet individual and a family man. So ever since the moon landing I think people have seen very little of him.
"It was a great privilege to spend a little bit of time with him."