It is hard to give credit to one single person. Here are some of the important people and their discoveries that led up to this model:
Ernest Rutherford found that the atom is mostly empty space with a very small but heavy nucleus in the center.
Niels Bohr had a model of atomic structure in which electrons orbited the nucleus in a circular orbit -- this was wrong (the orbits are not circular at all), but he had the idea that electrons were moving around the nucleus, which is true.
Werner Heisenberg was really the person that determined that the only way to describe the location of electrons is as a probability distribution, and this led to the idea of an "electron cloud" where you can only talk about the relative probability of find an electron in a certain volume. This is really what quantum mechanics is all about! Other important people involved in the development of quantum mechanics were Erwin Schrodinger and Louis de Broglie, among others.
See the Related Questions links for more information about each of these famous scientists!