hydrogen atom
If you wanted to learn how clocks work, you would start with the simplest one you could find — a single gear and a spring — before tackling a grandfather clock. The hydrogen atom is that simplest clock for atomic physics: one proton and one electron, nothing more. Its bare simplicity makes it the one atom we can solve exactly and learn the rules from.
The hydrogen atom is the lightest and simplest element's atom, consisting of a single positively charged proton with one electron bound to it. Because it has just one electron, the Schrödinger equation for it can be solved exactly, yielding clean orbitals, sharp energy levels, and a spectrum that the Rydberg formula predicts almost perfectly. It is the cornerstone on which our whole picture of atoms is built.
Why it matters: every heavier atom is understood by starting from the hydrogen solution and adding corrections for the extra electrons. The honest caveat is that hydrogen is uniquely simple precisely because it has only one electron — once a second electron is added, the exact solvability is lost and we must rely on approximations. Hydrogen also has heavier isotopes (deuterium, tritium) with the same single electron but extra neutrons.
When the electron in a hydrogen atom drops from the third energy level to the second, it emits red light at 656 nanometres — the bright red line you see in a hydrogen discharge tube. The wavelength matches the Rydberg formula to many decimal places, a triumph that helped launch quantum theory.
One proton, one electron — the only atom we can solve exactly.
A neutral hydrogen atom has no neutron in its commonest form (protium); its ion, H⁺, is simply a bare proton, while H⁻ carries an extra electron.