acceleration due to gravity
The acceleration due to gravity, written g, is the rate at which any freely falling object near Earth's surface speeds up as it falls: roughly 9.8 metres per second, gained every second, straight down. It is the single number that turns 'things fall' into something you can calculate. Because it is the same for all objects, it is a property of the place (Earth's surface) rather than of the falling object.
Precisely, g is about 9.8 m/s^2 near Earth's surface, directed toward the ground. In free-fall problems you simply set the acceleration a equal to g and use the constant-acceleration equations. There is a second face to g: it also equals the gravitational field strength, the force of gravity per kilogram of mass, which is why an object's weight is its mass times g, W = m g. So g in m/s^2 and g in newtons per kilogram are the same number describing the same thing from two angles.
The value 9.8 is not a universal constant — it is Earth-surface-specific and even varies a little from place to place: slightly larger at the poles, slightly smaller at the equator and up a mountain, because you are nearer or farther from Earth's centre. On the Moon g is about 1.6 m/s^2, so the same rock falls far more gently there. Do not confuse g (the local acceleration, about 9.8 m/s^2) with G, the universal gravitational constant that appears in Newton's law of gravitation — they are entirely different quantities.
A 2 kg bag has weight W = m g = 2 x 9.8 = 19.6 N on Earth. Carried to the Moon (g about 1.6 m/s^2) its mass is still 2 kg, but its weight drops to about 3.2 N — the same matter, a weaker local g.
g sets both how fast things fall and how much they weigh; it varies with location.
g (about 9.8 m/s^2, a local acceleration) is not G (6.67 x 10^-11, the universal gravitational constant). Same letter, different meaning: g depends on where you are, G is the same everywhere in the universe.