For the weather on Earth, its rotation makes a decisive contribution, the Coriolis force. As with the centrifugal force, it is only an apparent force, but on a non-rotating Earth, high and low pressure areas would look quite different.
Warm at the equator, cold at the poles
The solar radiation and thus the energy input per area is much greater at the equator than at the poles. Without the process "weather" it would become always hotter at the equator. But this leads to compensating currents, which reduce this temperature difference. Over the equator, the warmed air rises over a large area, and the air pressure on the ground drops. Over the subtropics the air sinks again, here a high pressure forms on the ground (Hadley cell). Over the polar regions lies heavy cold air, also here a high pressure forms at the ground. In between are the low pressure areas of the middle latitudes. On a non-rotating earth, primarily meridional circulation patterns (in north-south direction) would result.
The earth rotates
Now, however, the Earth is a rotating body that takes almost exactly 24 hours to make one revolution. At the equator the circumference is about 40 000 km, so a point located here moves with 1670 km/h in direction east. In the middle latitudes the circumference and also the rotation speed is smaller. Here in Switzerland, for example, we are moving eastward at just under 1000 km/h. If air masses now flow from the equator northwards, they take their higher speed with them and then move faster than the earth's surface. To an observer on the ground, it looks as if they are deflected from their northward course to the east –, i.e. to the right. Conversely, air parcels flowing southward from the pole are overtaken by the Earth's surface. So they are deflected on their south course to the west (again to the right).

Fig. 1: Schematic representation of the Coriolis force. Right deflection in the northern hemisphere, left deflection in the southern hemisphere.; Source: MeteoNews
This deflecting effect is the Coriolis force. It has its origin purely in the inertia of moving bodies. It has no effect on stationary objects. At the equator, the Coriolis force is zero, regardless of the speed of the object. At the same speed of the body, it rapidly increases in strength in the direction of the poles, especially at higher latitudes, and reaches its maximum at the pole.
Cyclones and anticyclones
Their contribution complicates the flow patterns on the rotating Earth! Land and sea are not evenly distributed and are heated differently (differential heating). Temperature differences lead to pressure differences, low (cyclones) and high pressure areas (anticyclones) are formed. Nature strives to balance temperature and pressure differences. Air flows from higher to lower pressure –, i.e. from a high pressure area into a low pressure area. On a non-rotating earth, it would do this on as direct a path as possible, but the Coriolis force deflects it in the process. This results in the formation of rotating high and low pressure areas. A depression rotates counterclockwise in the northern hemisphere and clockwise in the southern hemisphere. The trade winds blow from the subtropical high pressure belt towards the Innertropical Convergence Zone (ITC, low air pressure along the equator). They are also deflected accordingly. In the northern hemisphere the northeast trade wind blows, in the southern hemisphere the southeast trade wind.

Fig. 2: Schematic representation of the different circulation patterns on the rotating Earth.; Source: Wikipedia
The Coriolis force affects all moving objects, not only the air. It also has a decisive influence on ocean currents. Even in the field of technology, it can no longer be neglected in planning and construction from certain orders of magnitude. In the northern hemisphere, for example, the rails on the right in the direction of travel are subjected to somewhat greater stress in rail traffic.