How does the Coriolis effect influence global winds?

The Coriolis effect influences all winds, not just global ones.

Air moves from areas of high pressure to areas of low pressure. Sometimes this is on a local level like with a low pressure center, and sometimes it is on a global level, like how the air at the Equator has a higher pressure than air at the poles.

Regardless on what scale, the differences in pressure (air pressure gradient) cause the air to move. However, at the same time the Earth is rotating.

I am going to cut and paste part of an answer that I wrote on another question, as it is a really good way of explaining how Coriolis works.

Take a paper plate and put a marker in the middle. Pull the marker to the edge of the plate directly in front of you. You will get a straight line, from high pressure (center of the plate) to low pressure (edge). Now do it again only have a friend turn the plate while you do it. As you pull the marker directly toward you, the spot you were originally aiming for has moved. Once you reach the edge you will have a curved line, even though you moved the marker in a straight line. You have just demonstrated Coriolis effect.

In the Northern Hemisphere the Coriolis effect bends the winds in a counter clockwise direction. The opposite is true in the Southern Hemisphere.

I hope this is the answer you are looking for, otherwise please let me know.