Erwin Schrödinger looked at electrons as waves, each with its own wavefunction which is described by three parameters:
The energy of the orbital, defining whether it is low energy and close to the nucleus or high energy and further out.
The shape of the orbital; for example, it could be spherical or lobular; shaped like a dumbbell.
The magnetic moment of the orbital defined by its inclination.
Note that Wolfgang Pauli later added a fourth parameter, "spin", that has nothing to do with the classical concept of spin. Taken together, these parameters define the Quantum Numbers of a particle. Schrödinger disliked the idea of the discrete Quantum levels requiring a "quantum jump" from one level to the next. He hoped that a continuous wave theory would avoid it. Werner Heisenberg had no issues with Quantum jumps, and in the same year, 1925, that Schrödinger derived his wave equation, Heisenberg derived the Uncertainty Principle which states that it is possible precisely to measure either the position of a sub-atomic particle or its momentum, but not both.
Again, in 1925, Wolfgang Pauli proposed the "Pauli Exclusion Principle" which states: "There cannot exist an atom in such a quantum state that two electrons within it have the same set of quantum numbers". This means that electrons within the same orbital cannot have the same values for all four Quantum numbers. In the first orbital, closest to the nucleus, there are two electrons. By each having the opposite "spin", the Pauli Exclusion Principle is satisfied.
In 1928, Paul Dirac extended Pauli's work to take account of Special Relativity. His equation yielded possible negative values for the electron's energy, so he theorized that there would be an anti-electron. Carl Anderson discovered this particle in 1932, and called it a Positron.
In terms of our every day lives, probably the most profound aspect is "Quantum Tunneling". Consider an electron inside a box. Quantum mechanics tells us that there is a finite chance that the electron can disappear from inside the box and re-appear outside. This is Quantum tunneling. This is the basis of, for example, a transistor or a tunnel diode. It is also the mechanism for nuclear decay. Normally, protons and neutrons are bound inside of the nucleus of an atom. Alpha decay occurs when two protons and two neutrons (essentially a Helium nucleus) spontaneously tunnel out of the nucleus of a heavy element. Quantum tunneling tells us the probability of this happening. Enzymes use Quantum tunneling to enhance reaction rates by transferring electrons and certain atomic nuclei, for example hydrogen.
New Quantum Mechanics
Here is a really good, though somewhat mathematical, introduction to quantum mechanics.
This excellent site at North Carolina State University takes you on a non-mathematical journey from classical physics to the world of the quantum.