Short answers, pulled from the story.
A liquid is a state of matter with a definite volume but no fixed shape. It is nearly incompressible and adapts to the shape of its container in the direction of an applied force such as gravity. It is a form of condensed matter alongside solids and a form of fluid alongside gases.
Liquid is the least common state of matter in the known universe because liquids require a relatively narrow temperature and pressure range to exist. Most known matter is either gaseous, as in interstellar clouds, or plasma, as in stars.
Only mercury and bromine are liquid at standard conditions for temperature and pressure. Four more elements, francium, caesium, gallium, and rubidium, have melting points slightly above room temperature.
Viscosity is the measure of a liquid's resistance to flow, or more technically its resistance to deformation at a given rate. The viscosity of liquids decreases as temperature increases. Newtonian liquids like water and honey keep a constant viscosity, while non-Newtonian liquids like ketchup and custard thicken or thin under shear.
Surface tension arises because a molecule at the surface bonds with other liquid molecules only on the inner side, creating a net force that pulls surface molecules inward. It is measured as energy per unit area in joules per square metre, and liquids with strong intermolecular forces have larger surface tensions.
Liquids are hard to model because there is no small parameter from which to build a systematic theory, unlike the ideal gas reference for gases or the perfect lattice reference for solids. In liquids the attractive intermolecular forces and entropic forces are comparable, and the binding energy between molecules is the same order of magnitude as the thermal energy.
Quantum effects become important for liquids at low temperatures and with small molecular mass, when the thermal de Broglie wavelength is comparable to the average distance between molecules. Hydrogen and helium are notable examples, and liquid helium does not become solid even at absolute zero under standard pressure.