The word arch has several meanings:

1. A curved structure that spans an opening, supporting the weight above it. Ancient Roman builders used arches to create strong bridges and aqueducts that still stand today. The curved shape distributes weight outward and downward, making arches stronger than flat beams. You see arches in doorways, windows, and famous structures like the Gateway Arch in St. Louis. When something is arched, it means it's curved in this shape, like an arched eyebrow or an arched back.
2. To form a curve. A cat arches its back when startled. A basketball player might arch the ball high over a defender's outstretched hands. A dancer arches her back in a graceful pose.
3. Having a playfully teasing or mischievous quality. An arch smile suggests someone knows something others don't, or finds something secretly amusing. An arch comment might gently mock something in a clever way. This meaning often describes a knowing, slightly superior attitude, like when someone speaks with arch sarcasm.