I'm surprised that no one has mentioned the Mandelbrot set yet, arguably the most famous new mathematical object of the last 30 years, at least among the general public. Benoit Mandelbrot discovered it in 1979 as a result of computer experiments. He says that when he first saw it he was so surprised by its appearance that he thought it must be the result of a computer malfunction. In his book Fractals and Chaos, Mandelbrot argues that his discovery of the Mandelbrot set contributed to the revival of experimental mathematics and led to a general change in the attitude of mathematicians to experiments in mathematics. On page 25 he writes:
"The culture of mathematics during the 1960s and 1970s
Within that culture the Mandelbrot set could not have been discovered. Hence its discovery marked a historical departure. Today -- but not yesterday -- only a minority among mathematicians would agree with the opinion due to someone who did not discover that set, that the study of M reflects "a rather infantile and somewhat dull mathematical sensibility" (Brooks 1989)."