What discovery did J.J. Thomson make which disproved earlier scientific theory that claimed that the atom was an "indivisible" particle?

Prior to J. J. Thomson's discovery, atoms were believed to be the smallest particle, the buildings blocks of matter, and hence indivisible particles. Thomson disproved this by discovering electrons, which are significantly smaller than the atom itself, and as a result are called sub-atomic particles (among other sub-atomic particles, like the proton and neutron).

Thomson discovered the electron through his cathode ray tube experiment. Cathode ray tubes consist of hollow sealed glass with a vacuum...

Prior to J. J. Thomson's discovery, atoms were believed to be the smallest particle, the buildings blocks of matter, and hence indivisible particles. Thomson disproved this by discovering electrons, which are significantly smaller than the atom itself, and as a result are called sub-atomic particles (among other sub-atomic particles, like the proton and neutron).

Thomson discovered the electron through his cathode ray tube experiment. Cathode ray tubes consist of hollow sealed glass with a vacuum inside. On one end is a filament and an electric source, and on the other a screen (this is basically how older big TVs work). 

In his experiments, Thomson subjected the rays to a magnetic field and an electric field to see how they responded. He observed rays were bent by both a magnetic field and an electric field, which means they are charged. The way the rays responded to an electric field suggested that whatever particle was in the rays (during that time no one knew what they were) was negative. Using this, he was also able to determine the mass-to-charge ratio of the particle. He did this by analyzing the extent to which the ray was bent in the presence of an external electrical field. 

The experiment found the existence of a negatively charged particle with a really high mass-to-charge ratio. This could mean two things: the mass is really low, or the charge is really high. It would turn out that the mass of the particle is low. This particle is what we now call the electron, and it is certainly a lot smaller than an atom, proving the atom is in fact divisible (in the sense that it is composed of smaller particles called sub-atomic particles).