How do mountains affect where rain falls on opposite sides?

Orographic lift is the main mechanism - when moist air hits a mountain range, it's forced upward, cools as it rises, and drops most of its moisture as rain on the windward side. By the time that air crosses the peak and descends on the leeward side, it's already wrung itself out, so the opposite side ends up much drier. This creates what's called a rain shadow effect.

The windward side gets hammered with precipitation because the air keeps rising and cooling continuously as it climbs. Once it crests the mountain and starts descending on the other side, the air warms up again - and warm air can hold more moisture, so it doesn't drop rain anymore. Instead you get deserts or semi-arid regions on the lee side even though the windward side might be lush and wet. You see this pattern everywhere: the Cascades in the Pacific Northwest, the Rockies, the Alps. The western slopes face the moist ocean air and get soaked; the eastern slopes are bone dry.

The effect gets more dramatic with higher mountains and stronger weather systems pushing moisture inland. A really tall range can create a massive disparity - one side green and forested, the other side brown scrubland just a few miles away. It's one of those things that shapes entire ecosystems and settlement patterns without most people realizing there's a specific name for what's happening.