Q: Why does putting spin on a ball change how it moves through the air?

Physicist: This question is especially perplexing after a first year physics course, where every question starts with “ignoring air resistance…”. There are a couple of different way to approach the answer, but I like this one best.

Way back in the day a dude named D’Alembert came up with the unsurprisingly named “D’Alembert’s paradox” which essentially says that in air and water an object traveling at subsonic speeds experiences neither drag nor lift. That’s obviously not the case which is why this is “D’Alembert’s paradox” and not “D’Alembert’s well known fact”.
The resolution is to take into account vorticity and the rotation of air flow around the object in question.
Rotation of air allows the air to flow faster over the top or the bottom of the ball. But a funny thing happens when air moves quickly; the transverse air pressure (pressure to the sides) drops. This effect is called the Bernoulli effect or, in certain cases, the Venturi effect.

If you combine the flow of air rotating around a ball, with the flow of air past a ball, then the result is a faster flow past one side and a slower flow past the other side. In this example there is less pressure on the top than the bottom.

So how can you induce air to rotate around a ball in the air? Base balls, golf balls, and especially tennis balls are designed to have rough surfaces that grab the air a little bit. So if the ball spins as it moves it’ll drag the air around it a little, which leads to air rotating around the ball, which makes the air on one side move faster than the air on the other, which leads to a lower pressure on one side compared to the other, which creates a net force, which pushes the ball around.
So if you want the ball to curve downward you make it spin as though it were rolling away from you. If you want it to curve up (well… Stay in the air longer, no Human can make a ball actually curve upward in flight) you’d want the ball to spin as though it were rolling toward you.  And of course, you can also make the ball veer left or right using the corresponding spin.