What is wave-particle duality really about?

[Mason_C]

The wave-particle duality concept has always confused me. How can something be both a particle and a wave? Is it actually both at the same time, or does it behave like one or the other depending on how we measure it? What experiments best demonstrate wave-particle duality? And how does this relate to quantum mechanics as a whole? I'm looking for a clear explanation that doesn't oversimplify but also doesn't require advanced physics knowledge.