1. Light exhibits dual wave-particle behaviour as does all matter. The photoelectric effect and the discovery of light having momentum led to this belief and the theory that energy is absorbed and emitted in finite 'chunks' called quanta. The wavelength of macroscopic material, however, has such a small wavelength that we still treat them with the laws of Newtonian mechanics rather than wave mechanics. Since electrons have a tiny mass, however, their wavelength becomes somewhat significant.
2. Werner Heisenberg and Erwin Schrödinger went one step further on this. Heisenberg said if we know the position of the electron outside the nucleus we have no idea what its momemtum is and vice versa. Schrödinger then took this into account and proposed a new model of the atom. This is often illustrated as an electron cloud, where the denser the cloud, the greater the probability is of finding an electron.
3. A natural consequence of this is that there is an inherent imprecision in nature. The physical world is just based on probabilty.
(That's my attempt at explaining it. I'm not shocked by the theory so don't assume I know what I'm talking about.)