Plum pudding model
The plum pudding model was an early (and incorrect) 20th century model of an atom. It was proposed by J.J. Thomson in 1904, after the discovery of the electron, but before the discovery of the atomic nucleus. During that time, scientists knew that there was a positive charge in the atom that balanced out the negative charges of the electrons, making the atom neutral, but they didn't know where the positive charge was coming from. Thomson's model showed an atom that had a positively charged medium, or space, with negatively charged electrons inside the medium. Soon after its proposal, the model was called a 'plum pudding' model because the positive medium was like a pudding, with electrons, or plums, inside.
Development into modern atomic model
Rutherford's model
In 1909, not long after Thomson's model was proposed, Hans Geiger and Ernest Marsden made an experiment with thin sheets of gold, to see if Thomson was right. Their professor, Ernest Rutherford, expected the results to prove Thomson correct, but the results were extremely different to what they were expecting. In 1911, Rutherford discovered that the positive charges come from tiny particles called protons, and that the protons were in a tiny center called the nucleus, and that the electrons were orbiting around the nucleus.
Bohr model
Rutherford's model was like planets orbiting a star: electrons flying around a nucleus. But electrons have charge, unlike planets, and have energy. In 1913, Niels Bohr added 'energy levels' to the atomic model. Energy levels are quantum states: you have to be in one, or you have to be in the other. To change states you need energy. If an electron gets hit by a photon (a particle that carries electromagnetic radiation) it will gain extra energy and go into a higher energy level, then it will jump back down to a lower energy level, releasing its contained energy. This new model was called the Bohr model or the Rutherford-Bohr model. This added a whole new bit of science: Quantum physics.
Quantum model
In 1926 Erwin Schrödinger used the idea that electrons acted as a wave, as well as a particle, known as a wave-particle duality. This added a whole new layer to the atomic model and quantum physics. With a particle, you can know where it is in space if you observe (look) at it. But with a wave, it is all over the place, so you can't define where exactly it is. This is known as quantum uncertainty. With an electron, you can only know the probability of it being in a place, because it is a wave as well as a particle.