Short answers, pulled from the story.
The four levels of protein structure are primary, secondary, tertiary, and quaternary. Primary structure is the sequence of amino acids; secondary structure includes local sub-structures such as alpha-helices and beta-sheets; tertiary structure is the full three-dimensional shape of a single polypeptide chain; quaternary structure describes assemblies of two or more chains into a functional unit.
Frederick Sanger determined the amino acid sequence of insulin, establishing that proteins have defining amino acid sequences unique to each protein. Insulin is composed of 51 amino acids arranged in two chains: one chain has 31 amino acids and the other has 20.
Protein folding is driven by a combination of hydrophobic interactions, hydrogen bonding, van der Waals forces, and Coulomb interactions. Hydrophobic collapse pushes hydrophobic side-chains into the protein's core and away from water, while hydrophilic side-chains face outward toward the solvent.
Anfinsen's dogma is the thermodynamic principle that a protein's native three-dimensional structure is uniquely determined by its amino acid sequence. It states that the native structure represents a unique, stable, and kinetically accessible minimum of free energy.
Around 90% of structures in the Protein Data Bank have been determined by X-ray crystallography, which maps the electron density of a crystallized protein to infer atomic coordinates. Nuclear magnetic resonance techniques account for roughly 7% of known structures, and cryo-electron microscopy is used for very large complexes such as virus coat proteins and amyloid fibers.
Intrinsically disordered proteins exist and function in a state that lacks a stable tertiary structure, making them impossible to describe with a single fixed three-dimensional form. Conformational ensembles are used to represent their behavior; examples include Sic1/Cdc4, p15 PAF, MKK7, Beta-synuclein, and P27.