A wave is a traveling disturbance
Watch a crowd do 'the wave' in a stadium: a ripple of standing-and-sitting sweeps around the ring, yet no single person runs around the stadium. Drop a pebble in a pond and rings race outward, but a floating leaf just bobs up and down — it is not carried to shore. Pluck a guitar string and a shiver runs down its length. In every case a pattern travels while the material mostly stays put. That is the essence of a wave.
A wave that needs a material to travel through is a mechanical wave: sound needs air, water waves need water, a string wave needs the string. Take the medium away and the wave dies. (Light is the great exception — an electromagnetic wave sails through empty space with no medium at all. We meet that later.)
Two ways to wiggle: transverse and longitudinal
In a transverse wave the medium moves perpendicular to the direction the wave travels. Shake a rope side-to-side and the wave runs along the rope while each bit of rope moves across it. Water-surface ripples and light are (largely) transverse.
In a longitudinal wave the medium moves parallel to the travel direction — back and forth along the same line. Push and pull the end of a slinky and a squeeze runs along its length. Sound is the great example: air is alternately compressed and stretched. Same idea, different geometry.
Anatomy of a wave
Four numbers describe a simple wave. The wavelength \lambda is the distance between two successive crests — the wave's repeat length in space. The amplitude A is the maximum displacement from rest; it sets how much energy the wave carries. The period T is the time for one full oscillation, and the frequency f is how many oscillations happen per second (in hertz, Hz).
Period and frequency are reciprocals: 2 Hz means two cycles per second, so each cycle lasts half a second.
The master equation: v = fλ
Here is the single most useful fact about waves. In exactly one period T, the wave advances exactly one wavelength \lambda. So its wave speed is the distance \lambda divided by the time T — which, since f = 1/T, equals f\lambda.
Wave speed = frequency × wavelength. It links the three quantities and is the workhorse of the whole subject.
Quick example. A wave on a lake has wavelength \lambda = 2\text{ m} and frequency f = 3\text{ Hz}. Then v = f\lambda = 3 \times 2 = 6\text{ m/s}. Every crest reaches you 6 metres closer each second — with three crests arriving per second.