Strictly speaking, the concept of pressure gradient is a local characterization of the air (more generally of the fluid under investigation). The pressure gradient is defined only at those spatial scales at which pressure (more generally fluid dynamics) itself is defined.
Within planetary atmospheres (including the Earth's), the pressure gradient is a vector pointing roughly downwards, because the pressure changes most rapidly vertically, increasing downwards. The value of the strength (or norm) of the pressure gradient in the troposphere is typically of the order 9 Pa/m (or 90 hPa/km).
The pressure gradient often has a small but critical horizontal component, which is largely responsible for the wind circulation. The horizontal pressure gradient is a 2-dimensional vector resulting from the projection of the pressure gradient onto a local horizontal plane.
Near the Earth's surface, this horizontal pressure gradient is typically pointing towards high pressure air masses (anticyclones), its particular orientation at any one time and place depends strongly on the weather situation. At mid-latitudes, the typical horizontal pressure gradient may take on values of the order of 10-2 Pa/m (or 10 Pa/km), although rather higher values occur within meteorological fronts.
Differences in air pressure between different locations are critical in weather forecasting and climate. As indicated above, the pressure gradient constitutes one of the main forces acting on the air to make it move as wind. Note that the pressure gradient force points from high towards low pressure zones, it is thus oriented in the opposite direction from the pressure gradient itself.
Our ears detect and interpret sound waves; as such they are delicate, sensitive organs to measure pressure gradients. Excessive pressure gradients can hurt or damage the auditory system. By contrast, pressure alone does not have quite the same effect: divers can reach significant depths under water (and thus experience very large pressures) without damage, provided appropriate and necessary precautions are taken.