John Dalton is the first scientist to take Democritus' philosophy of atoms and devise experiments to try to prove that atoms exist. Dalton's experiments at the turn of the 19th century involved evaporating water and discovering that it existed in the air as a gas with the exact same mass as it did when it was in liquid form. From this experiment, Dalton concluded that water must be made up of individual particles that retain their mass no matter the state the water was in, and thus modern atomic structure was born.
The first elementary particle discovered in atoms was the electron. Near the end of the 19th century, J.J. Thomson used a cathode ray tube --- a tube that contains a small amount of gas between two metal plates --- to show that all atoms were composed of individually charged negative particles, which he called "electrons." This proof came because all atoms, regardless of the type of element, placed in the cathode ray reacted the same way, which meant that all atoms had to be made up of the same elementary particles.
Thomson postulated that an atom had an equal amount of electrons and unknown positive particles, and that they were evenly distributed in the atom. This model of atomic structure --- known as the "plum pudding model" --- was proven false by Ernest Rutherford's 1909 gold sheet experiment. Rutherford shot atoms into a thin sheet of gold paper and discovered that some deflected off of it by a huge degree. Rutherford concluded that this deflection was caused by a large positive mass in the center of the nucleus; he later coined the term "proton" to describe these particles that make up the nucleus of the atom.
By the 1930s, the current model of atomic structure --- electrons orbiting protons --- was established. However, the work of James Chadwick showed that there was a third particle in the atom --- the "neutron." Chadwick performed experiments that showed radiation released a particle from atoms that was not affected by electrical fields, and thus had to be electrically neutral. Neutrons were found to be in the nucleus of the atom alongside protons.