Short answers, pulled from the story.
An atomic orbital is a mathematical function describing the location and wave-like behavior of an electron within an atom. Erwin Schrödinger published this equation in 1926 to describe electrons as standing waves rather than tiny planets orbiting a nucleus.
Hantaro Nagaoka published an orbit-based hypothesis for electron behavior as early as 1904 with his Saturnian Model. This model placed positive charge into a central core pulling electrons into circular orbits reminiscent of Saturn's rings.
Each orbital in an atom is characterized by three quantum numbers: n, l, and m_l. These correspond respectively to an electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis.
Single s orbitals (l=0) are shaped like spheres because they lack angular momentum and avoid having a wave node at the nucleus. The three p orbitals for l=1 have the form of two ellipsoids with a point of tangency at the nucleus due to their non-zero angular momentum.
The filling of the 3d orbitals does not occur until the 4s orbitals have been filled according to the energy sequence. This specific order of filling is responsible for the structure of the periodic table and explains why elements in the same block share common features.