φ (phi) ≈ 1.618 is also the value that the ratio of two consecutive Fibonacci numbers (1,1,2,3,5,8,13…) converges to. Keep dividing a golden rectangle into a square plus a smaller golden rectangle, connect the squares' arcs, and you get the famous golden spiral.
In web layout it shows up as the width ratio between two columns (say, content 61.8% : sidebar 38.2%), an image-to-caption ratio, or the height ratio between a hero section and the content below it. You can also apply it directly with aspect-ratio: 1.618.
The "it's beautiful because it's the golden ratio" claim is often overstated, though — in practice any non-extreme asymmetric ratio (roughly 1.3 to 1.8) tends to look fine. It's more useful as a rule of thumb — "an exact 50/50 split reads as boring" — than as a number to worship.
When to use
Use a ratio near 61.8:38.2 as a reference when you want two areas to feel subtly unequal in weight. The clearly-asymmetric feel matters more than hitting the exact number.