Short answers, pulled from the story.
The old quantum theory is a collection of results from the years 1900-1925 that predate modern quantum mechanics. It was a set of heuristic corrections to classical mechanics, never complete or self-consistent, and is now understood as the semiclassical approximation to modern quantum mechanics.
Max Planck instigated the old quantum theory with his 1900 work on the emission and absorption of light in a black body, introducing his quantum of action. Albert Einstein's 1907 work on the specific heats of solids brought the approach to wider attention and marked the beginning of the theory in earnest.
The main final accomplishments of the old quantum theory were the determination of the modern form of the periodic table by Edmund Stoner and the Pauli exclusion principle, both premised on Arnold Sommerfeld's enhancements to the Bohr model of the atom.
The Bohr-Sommerfeld model extended Bohr's 1913 hydrogen atom model by allowing electron orbits to be ellipses instead of circles and introducing quantum degeneracy. Sommerfeld quantized the z-component of angular momentum, a phenomenon called space quantization, bringing the model closer to the modern quantum mechanical picture than Bohr's original version.
The old quantum theory could not calculate spectral line intensities, failed on the anomalous Zeeman effect involving electron spin, and could not quantize chaotic systems. It was replaced in 1925-1926 by Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation, with Paul Dirac proving in 1926 that both methods follow from a more general transformation theory.
In 1924, as a PhD candidate, Louis de Broglie proposed that all matter, electrons as well as photons, is described by waves. He reinterpreted the old quantum condition as a requirement for constructive interference, explaining why orbits are quantized: matter waves form standing waves only at discrete energies.