For a long time the atom was believed to be the smallest, indivisible unit of matter — Dalton's word atomos literally means "uncuttable". Experiments with electricity in the late nineteenth century shattered that picture by revealing that atoms themselves are built from still smaller sub-atomic particles: the electron, the proton and the neutron.
The electron was discovered through cathode ray experiments in a discharge tube. J. J. Thomson showed that these rays travel from cathode to anode, are deflected by both electric and magnetic fields towards the positive plate, and behave identically whatever gas or electrode is used — so a negatively charged particle is common to all matter. By balancing the electric and magnetic deflections he measured the charge-to-mass ratio $\frac{e}{m_e}=1.758\times10^{11}\,\text{C kg}^{-1}$. R. A. Millikan's oil-drop experiment later fixed the electron's charge as $e=1.602\times10^{-19}\,\text{C}$, giving the electron mass $m_e=9.11\times10^{-31}\,\text{kg}$.
Positively charged canal rays (anode rays) revealed the proton. Unlike cathode rays, their charge-to-mass ratio depends on the gas, and it is largest for hydrogen — so the hydrogen positive ion (the proton) is the fundamental unit of positive charge, with charge $+1.602\times10^{-19}\,\text{C}$ and mass $1.672\times10^{-27}\,\text{kg}$ (about $1837$ times the electron). The electrically neutral neutron, of nearly the same mass, was discovered by James Chadwick in 1932 when he bombarded beryllium with $\alpha$-particles.
- Atomic number $Z$ = number of protons = number of electrons in a neutral atom; it fixes the element's identity.
- Mass number $A$ = protons + neutrons (nucleons). Number of neutrons $= A-Z$.
- Isotopes: same $Z$, different $A$ (e.g. $^{1}_{1}\text{H}$, $^{2}_{1}\text{H}$, $^{3}_{1}\text{H}$). Isobars: same $A$, different $Z$ (e.g. $^{40}_{18}\text{Ar}$ and $^{40}_{20}\text{Ca}$).
Thomson's model (the "plum-pudding" model) pictured the atom as a uniform sphere of positive charge with electrons embedded in it like seeds in a watermelon. It explained electrical neutrality but failed completely when Geiger and Marsden fired $\alpha$-particles at thin gold foil. Most particles passed straight through, a few were deflected at large angles, and roughly $1$ in $20000$ bounced almost straight back. From this Rutherford concluded that the atom is mostly empty space, with a tiny, dense, positively charged nucleus (radius $\sim10^{-15}\,\text{m}$ against an atomic radius $\sim10^{-10}\,\text{m}$) around which electrons orbit. But classical physics says an accelerating (orbiting) electron must radiate energy and spiral into the nucleus — Rutherford's atom should collapse in $\sim10^{-8}\,\text{s}$, and it could not explain line spectra.
Bohr's model rescued the hydrogen atom with three postulates: the electron moves only in certain stationary orbits without radiating; angular momentum is quantised, $mvr=\frac{nh}{2\pi}$ (with $n=1,2,3,\dots$); and energy is absorbed or emitted only when the electron jumps between orbits, $\Delta E=E_{\text{final}}-E_{\text{initial}}=h\nu$. For hydrogen the allowed energies and radii come out as $E_n=-\frac{13.6}{n^2}\,\text{eV}$ and $r_n=0.529\,n^2\,\text{angstrom}$, so $r_n\propto n^2$. The negative sign means the electron is bound; $13.6\,\text{eV}$ is the ionisation energy of hydrogen.
Bohr's model beautifully reproduces the hydrogen spectrum. When excited electrons fall back, they emit photons of fixed wavelength, grouped into series. The Rydberg equation gives the wavenumber of every line: $\bar{\nu}=\frac{1}{\lambda}=R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$, with $R_H=1.097\times10^{7}\,\text{m}^{-1}$ and $n_2>n_1$. The Lyman series ($n_1=1$, ultraviolet), Balmer series ($n_1=2$, visible) and Paschen, Brackett, Pfund series ($n_1=3,4,5$, infrared) all follow from one formula. Its limitations: it works only for one-electron species (H, $\text{He}^+$, $\text{Li}^{2+}$), cannot explain fine spectral structure or the splitting of lines in magnetic and electric fields (Zeeman and Stark effects), ignores the wave nature of the electron, and violates the uncertainty principle by assigning a definite orbit.