Mass and Inertia
Mass is the amount of matter in an object and the measure of its inertia; more mass means more resistance to changes in motion.
Definition
Mass is how much stuff an object is made of, measured in kilograms, and the more mass something has, the more inertia it has, so it is harder to start moving or to stop. Mass is the quantitative measure of inertia and does not change with location: an object has the same mass on Earth, on the Moon, or in deep space, while weight, the force of gravity on that mass, does change. Inertial mass is defined operationally by Newton's second law, $m = |\vec{F}|/|\vec{a}|$, while gravitational mass is defined by the gravitational force law $F = GMm/r^2$. Their measured equality, the equivalence principle, explains why free-fall acceleration is independent of mass. The kilogram is now defined through the Planck constant, tying mass to fundamental constants rather than a physical artifact.
Example
It is easy to stop a rolling tennis ball with your foot, but stopping a rolling bowling ball takes much more effort, because the bowling ball has more mass and more inertia. A $10\ \text{kg}$ box weighs $98\ \text{N}$ on Earth but only about $16\ \text{N}$ on the Moon, yet its mass is still $10\ \text{kg}$ in both places, and a $20\ \text{N}$ push gives it $2\ \text{m/s}^2$ on either world. Mass can even be measured purely from inertia. When cart A of unknown mass at $3\ \text{m/s}$ hits stationary cart B ($2\ \text{kg}$) on a frictionless track and they stick together moving at $1\ \text{m/s}$, momentum conservation gives $m_A \times 3 = (m_A + 2) \times 1$, so $m_A = 1\ \text{kg}$.
Key Insight
Mass is not the same as size. A balloon can be bigger than a brick but have far less mass, which is why the brick is much harder to throw. A balance compares masses and gives the same reading anywhere, while a spring scale measures weight and reads differently on the Moon; that difference is the whole distinction between mass and weight. In special relativity, the energy needed to accelerate an object grows without bound as it approaches $c$, so inertia effectively increases with speed. Rest mass is the invariant quantity, the same in every reference frame.