Which effect makes a spinning ball curve sideways through the air?

The Magnus effect. A spinning ball drags air around with it, so the air on one side moves faster relative to the oncoming flow and the other slower; the pressure differs across the ball and it is pushed sideways. Every curved free kick, sliced drive and topspin forehand is this.

Newton described it from tennis in 1672, well before Magnus, who investigated it in 1852 because artillery shells were drifting off target. The size of the force depends on spin rate, speed and the surface - which is why a cricket ball is polished on one side and roughened on the other, and why the dimples on a golf ball matter enormously: they hold a turbulent layer against the surface that both cuts drag and increases the lift backspin generates.

It reverses under some conditions, which is what makes a knuckleball unpredictable and why football manufacturers get into trouble. The 2010 World Cup's Jabulani had an unusually smooth surface, so the airflow switched between smooth and turbulent regimes at speeds common in the game, and the ball genuinely did behave erratically - goalkeepers were not imagining it, and wind tunnel studies afterwards agreed with them.