Gravity
Gravity is the attractive force between any two masses; it grows with mass and weakens with the square of the distance between them.
Formula
F = G\frac{m_1 m_2}{r^2}
Definition
Gravity is the pull that draws objects toward each other; Earth's gravity pulls everything toward the ground and keeps the Moon circling around us. It is a non-contact force of attraction between any two objects that have mass, growing stronger as the masses increase and weaker as the distance between them increases, and near Earth's surface it gives every falling object the same acceleration, $g = 9.8\ \text{m/s}^2$, ignoring air resistance. Newton's law of universal gravitation gives the force between point masses as $F = Gm_1m_2/r^2$, with $G = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$, acting along the line joining the masses and obeying the inverse-square law. Near a planet's surface it reduces to $F = mg$ with $g = GM/R^2$. Gravity is a conservative force, so it defines a potential energy $U = -Gm_1m_2/r$.
Example
Drop a pencil and it falls to the floor; throw a ball up and it comes back down, because gravity pulls both toward Earth's center. A rock dropped from a cliff and falling for $3\ \text{s}$ reaches a speed of $v = gt = 9.8 \times 3 = 29.4\ \text{m/s}$ and has fallen $d = \tfrac{1}{2}gt^2 = 0.5 \times 9.8 \times 9 = 44.1\ \text{m}$. Computing Earth's surface gravity from the law itself, $g = GM/R^2 = (6.67 \times 10^{-11})(5.97 \times 10^{24})/(6.37 \times 10^6)^2 \approx 9.8\ \text{m/s}^2$. At the altitude of the International Space Station, $400\ \text{km}$ up, $g$ drops only to about $8.7\ \text{m/s}^2$, so astronauts float because they are in free fall, not because gravity is absent.
Key Insight
Everything with mass has gravity, even you; your pull is just far too weak to notice because your mass is tiny compared to a planet. A feather and a hammer dropped together on the Moon land at the same time, and on Earth the feather falls slower only because of air resistance, not because gravity pulls it less strongly per kilogram. Gravity is by far the weakest of the fundamental forces, yet it dominates the universe because it is always attractive and reaches infinitely far. Electric charges cancel out on large scales; mass never does.