As with most great artists from over the millennium, Michelangelo di Lodovico Buonarroti Simoni, known simply as Michelangelo, is remembered for his most famous works of art, especially for his magnificent paintings that adorn the ceiling of the Sistine Chapel in the Vatican, paintings commissioned by Pope Julius II. Michelangelo, together with Leonardo Da Vinci, is considered one of the seminal figures of the Renaissance period in European history, and he remains synonymous with the flowering of artistry and openness of expression that defined that era. Michelangelo's paintings and sculptures reveal a commitment to realism combined with the imagery associated with religious orthodoxy. His famous depiction in the Sistine Chapel collage of God and Adam extending outstretched arms towards each other, formalizing the relationship between the Divine Being and His first human creation, has become so iconic an image that it continues to be reproduced on all-manner of commercially-available objects, including coffee mugs, pillows, and t-shirts.
More seriously than the exploitation of great works of art for commercial purposes, Michelangelo's relevance for the current era can also be found in his enduring influence on architectural design, such as the relevance of his design for the Laurentian Library in Florence, which remains a marvel of design hundreds of years after its completion. His work for the powerful Medici, including the chapel and tombs for members of that family, continue to be studied for their designs, especially the tomb for Lorenzo de Medici, with its carefully considered depictions of imagery representative of the deceased's character.
Michelangelo remains relevant today because of the beauty of his creations and, as importantly, for the social, political and religious contexts in which he operated. The Renaissance era was enlightening, but also dangerous, the power of the church over all facets of society during those years. That Michelangelo succeeded, despite occasional conflicts with his powerful, religious overseers, in bridging the divide between theological orthodoxy and piety to the reigning powers-that-be remains an achievement of enormous magnitude.
Your question is in the singular, but it really should be plural, because Michelangelos's contribution to our society is nothing less than great. He has inspired generations of artists and people with his amazing paintings and sculptures. For example, he has painted arguably one the most recognized painting in the world on the Sistine chapel - a painting of the history of western art and a scene from Genesis. He also carved out the famous statue of David. Also he pushed the limits of architecture as he design one of the most amazing domes of the world - the dome of Saint Peter's church. In short, he has shown the world what a talented person can accomplish.