Turning the question around
Light shows two faces. Interference and diffraction are wave behaviour; the photoelectric effect is particle behaviour. In 1924 Louis de Broglie asked the audacious symmetric question: if a light wave carries particle-like momentum, perhaps a particle of matter — an electron — carries a wavelength.
The de Broglie wavelength: every moving object has one, set by its momentum.
The clue came from light. A photon has momentum p = E/c = hf/c = h/\lambda. De Broglie simply turned this around, \lambda = h/p, and insisted it applies to everything with momentum — electrons, atoms, baseballs. This is the de Broglie wavelength.
Electrons that interfere
This was no idle guess. Davisson and Germer fired electrons at a nickel crystal and got a diffraction pattern, just like X-rays — the crystal's atomic rows acted as a grating for the electron waves. Fire electrons through a double slit one at a time, and each arrives as a single dot, yet over thousands of hits the dots pile up into interference fringes. Each electron interferes with itself.
Why don't we notice matter waves? Because h is minute. An electron moving at 10^6 m/s has \lambda \approx 0.7 nm — the size of atoms, so it diffracts off crystals and is the basis of the electron microscope. A 0.145 kg baseball at 40 m/s has \lambda \approx 10^{-34} m, unimaginably smaller than any nucleus and utterly unmeasurable. Matter waves are always there; for big things they are hopelessly tiny.
The uncertainty principle
Waves force a trade-off. A wave stretched out smoothly across space has one clean wavelength — hence a sharp momentum — but no definite position. To pin a wave to one spot you must add up many wavelengths into a compact wave packet, which blurs the momentum. You cannot make both position and momentum sharp at once.
Heisenberg's uncertainty principle: the spreads in position and momentum have a floor.
The consequences are real. It is why an electron cannot sit still trapped inside the tiny nucleus (squeezing \Delta x small forces a huge \Delta p and energy), why atoms have a lowest 'zero-point' energy they can never lose, and why the crisp determinism of Newton's world softens into probability at the quantum scale.
What duality really means
It is tempting to picture an electron as 'sometimes a wave, sometimes a particle,' flipping between costumes. That is not quite it. A quantum object is its own kind of thing; which face it shows depends on the question you ask it. A which-path experiment reveals particles; an interference experiment reveals waves — and you cannot see both sharply in the same run. Bohr called this complementarity.
With wave–particle duality and uncertainty in hand, the full machinery of quantum mechanics — the wavefunction, the Schrödinger equation, probability clouds — takes over, and that is the subject of a dedicated domain. But we already have enough to do something spectacular: explain the atom and the colours it emits. That is next.