He appeared on the BBC One show from 2010 until 2013, partnering with Patsy Kensit, Anita Dobson, Lisa Riley and Deborah Meaden.
Strictly Come Dancing co-host Claudia Winkleman described the news of Windsor's death as "heartbreaking".
Presenter Tess Daly said it was "such terribly sad news", adding that Windsor was "gone too young".
"Your infectious personality, energy and talent will be deeply missed by all who knew you," Daly said. "My heart goes out to your loved ones at this time."
The cause of Windsor's death has not been confirmed.
Good Morning Britain presenter Susanna Reid became emotional as she announced Windsor's death on Tuesday.
"He was a remarkable dancer, incredibly strong, so creative, he was just so enthusiastic about dancing," she said. "I just adored him, he was the most incredible personality, there was something very special about him."
Windsor was a 'brilliant dancer and a lovely man'
Windsor danced with Reid on a Children in Need special of Strictly. He was also paired with Great British Bake Off host Alison Hammond in the 2015 Christmas special.
Strictly's longest-serving judge Craig Revel Horwood described Windsor as "one of the kindest, gentle, honest, funny and caring people I've had the pleasure of knowing and working with".
"He will be forever missed. My heart goes out to all his family and friends."
Strictly's head judge Shirley Ballas added: "I'm very saddened to hear the devastating news about this beautiful human being this morning.
"Robin Windsor… kind, gracious, gentle and extremely hard working. A wonderful dancer, true gentleman, good soul and so very young."
Former judge Bruno Tonioli added: "Incredibly sad shocking news. It was a pleasure working with Robin for many years at Strictly, so professional easy going a really good person."
In a statement, a Strictly Come Dancing spokesperson said: "The whole Strictly Come Dancing family are deeply saddened to hear the news about our dear friend Robin Windsor.
"He was not only an exceptionally talented dancer and choreographer but also a caring, considerate and kind person both on and off the dancefloor. Our thoughts are with his friends and family at this extremely difficult time."
Actress Lisa Riley, who was paired with Windsor in 2012, posted: "My Bestie, My Robin, My Angel... now our forever Angel, who is loved, will always be loved, forever in my heart.
"I love you, shine your beautiful, electric energy from heaven."
In a statement read out on This Morning, Hammond said: "I'm completely heartbroken. He was one of the nicest people to work with and so much fun to be around.
"I had the pleasure of dancing with him for the Strictly Christmas special and they were such special times. My thoughts are with his family and friends."
Announcing Windsor's death, Sisco Entertainment told BBC News: "His extraordinary ability to express emotion through movement captivated audiences, leaving them spellbound with every step.
"Beyond his impeccable dancing, Robin's vibrant spirit lit up the room, infusing rehearsals and performances with infectious energy and warmth.
"His laughter was contagious, his kindness boundless. His friendship was a gift cherished by all who had the privilege of knowing him. Robin, you will be deeply missed."
Former Strictly pro James Jordan said: "I had the pleasure of working with this man for many years on Strictly - I have such fond memories of him. He was always very caring and fun to be around.
"Everyone who came in contact with him adored him, from his celebrity partners to his working colleagues. You will be missed old friend."
Ballroom dance company Burn The Floor said it had "lost one of its founding members".
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