Wave
A wave is a disturbance that carries energy from one place to another without carrying matter along with it.
Definition
A wave is a wiggle that travels and carries energy from one place to another while the medium it moves through wobbles but stays where it is: waves move energy, not material, the way a stadium wave travels around the stands while every fan stays in their own seat. Mechanical waves such as sound, water, and seismic waves need a medium, while electromagnetic waves such as light and radio can travel through a vacuum, and every wave is described by wavelength, frequency, amplitude, and speed, related by $v = f\lambda$, with the medium moving perpendicular to the wave direction in a transverse wave and parallel to it in a longitudinal wave. Mathematically a wave solves the wave equation $\partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2$, and a traveling sinusoidal wave takes the form $y(x,t) = A\sin(kx - \omega t)$, with wave number $k = 2\pi/\lambda$ and angular frequency $\omega = 2\pi f$, so $v = \omega/k = f\lambda$; wave speed is set by the medium, $v = \sqrt{T/\mu}$ for a string, $v = \sqrt{B/\rho}$ for a fluid, and $c = 1/\sqrt{\mu_0\varepsilon_0}$ for electromagnetic waves, and waves obey superposition, producing interference, diffraction, and standing waves, with energy carried proportional to amplitude squared.
Example
Drop a pebble in a pond and ripples spread out while a nearby leaf bobs up and down without traveling with them; ocean waves arriving every $8\ \text{s}$ with crests $40\ \text{m}$ apart have frequency $1/8 = 0.125\ \text{Hz}$ and speed $v = f\lambda = 0.125 \times 40 = 5\ \text{m/s}$. A guitar string of length $0.65\ \text{m}$, mass $3.0 \times 10^{-3}\ \text{kg}$, under $80\ \text{N}$ tension has $\mu = 4.6 \times 10^{-3}\ \text{kg/m}$ and $v = \sqrt{80/4.6 \times 10^{-3}} \approx 132\ \text{m/s}$, so its fundamental with $\lambda = 2L = 1.3\ \text{m}$ has $f = v/\lambda \approx 101\ \text{Hz}$.
Key Insight
Sound cannot travel through space because there is no air to carry it, while light can, because electromagnetic waves are self-supporting and need no medium. Two waves can pass through each other and emerge unchanged, which particles cannot do, and that superposition principle is what makes noise-cancelling headphones and holograms possible.