The de Broglie wavelength is the wavelength associated with what we call matter waves - i.e. waves associated with particles that are in motion. Now, do keep in mind that these matter waves are starkly different from waves on a string, or a water surface: these are probabilistic waves that indicate how likely it is to locate a particle at a particular point in space and time. The concept of de Broglie wavelength is extremely relevant in the world of modern physics, because it is considered to be the cornerstone of quantum mechanics - the one field of physics that changed the status quo in the early 20th century.
Before quantum mechanics had ever entered the limelight, physicists were under the impression that everything in this world that needed discovery had already been dealt with. But it was through phenomena like emission/absorption line spectra and photoelectric effect that something extremely strange was proven: a wave like electromagnetic waves show particle behavior. This had been explained by the idea that energy must be quantised, not continuous, which ultimately opened the gates for quantum mechanics to flourish.
From these experiments then, it was natural to suggest the inverse - that perhaps particles can also behave like waves!
This spooky blurring of the wave and particle was explained most beautifully in what we call the wave-particle duality: that physical entities have the ability to exhibit either particle or wave properties depending on the type of observation being made. De Broglie took this a step further and suggested that if a wave like light can have an ascribed wavelength, then surely particles’ matter waves must have their own de Broglie wavelength!
The way he derived it was as follows:
Upon connecting these equations, we get:
Now, here de Broglie waved his hand a little! Because this equation is being related to waves associated with particles, one must recall that particles cannot travel at the speed of light c. Therefore the constant c must be replaced with another dimensionally consistent term: velocity of the particle v. This makes more sense, because now the equation can be further simplified:
