Although the magnitude of the acceleration due to gravity, g, is the same everywhere on the Earth's surface, its direction changes depending on where you are. It is a vector that always points towards the center of the Earth, so, for example, it is in the opposite direction at the North Pole than at the South Pole. This effect is not very relevant to us because the Earth is so big. If we move from one end of the room to another, or even one end of the city to another, we are only moving across a very small fraction of the total circumference of the Earth, so the direction of the gravitational acceleration changes very little. Our notion of ``down'' only changes significantly when we travel very large distances. However, as we will see later, if you happen to be near a very massive, but small object, such as a black hole, the fact that gravitational acceleration changes direction depending on your location becomes very significant indeed: it gives rise to so-called tidal gravitational forces that can tear a spaceship apart in microseconds.
Note that the rate at which objects accelerates towards the Earth changes with distance. We don't notice this because we are already so far from the center of the Earth (6 thousand kilometers) that we would have to move vertically a large distance to cause a significant change: for example, to decrease the acceleration due to gravity by half down to 4.9 m/s2 we would have to go out into space about 1500 km above the Earth's surface.