Short answers, pulled from the story.
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds. Depending on context, the term may or may not include ions that satisfy the same criterion, and in quantum physics, organic chemistry, and biochemistry the distinction from ions is dropped.
Amedeo Avogadro created the word molecule. In his 1811 paper, he argued that the smallest particles of gases are not necessarily simple atoms but are made of a number of atoms united by attraction to form a single molecule.
The French physicist Jean Perrin proved that molecules exist, and he received the Nobel Prize in physics for it in 1926. He calculated the Avogadro constant using three different methods involving liquid phase systems, including a gamboge emulsion and experimental work on Brownian motion.
The smallest molecule is diatomic hydrogen, H2, with a bond length of 0.74 angstroms. The hydrogen molecule-ion, H2+, has the simplest of all chemical bonds, a single one-electron bond.
Metals and salts are not made of discrete molecules because they form large crystalline networks or repeating unit cells held together by chemical bonds. Solid metals are condensed phases with metallic bonding, and no typical molecule can be defined for salts or covalent crystals such as diamond, quartz, and sodium chloride.
Covalent bonding involves the sharing of electron pairs between atoms and holds most molecules together. Ionic bonding involves the electrostatic attraction between oppositely charged ions, where atoms that lose electrons become cations and atoms that gain electrons become anions, a transfer termed electrovalence.
An empirical formula gives the simplest integer ratio of the elements in a compound, while a molecular formula reflects the exact number of atoms. For example, acetylene has the molecular formula C2H2 but the simplest integer ratio CH.