For a periodic wave, the wavelength is the distance after which the wave repeats its shape1. The wavelength is usually denoted with the Greek letter λ and has dimensions of length. For a perfect sinusoidal wave, the wavelength can be measured as the distance between one crest and the next or one trough and the next. If a wave has a long wavelength, it changes very slowly over distances. If the wavelength is small then it changes very rapidly over distances. The wavelength describes the behavior of a wave in space in the same way that the period describes the behavior in time. If a traveling wave varies in time with a period T, then the speed of the wave will be λ/T.
Mathematically, a sinusoidal wave takes on values according to the expression
at a position x. This expresses the fact that the value for x = 0 will be the same as for x = λ, since that would just add 2π to the argument of cosine, which does not change the value. It is true that the value is the same for any two points separated by a distance λ. Often when dealing with the mathematics of waves it is more convenient to describe the behavior in terms of wavenumber rather than wavelength. In general, a wave that is not perfectly sinusoidal is made up of a superposition (the addition of) many sinusoidal waves of different wavelengths.
As it characterizes a wave, the wavelength will determine how many other wave phenomina exhibit themselves, including: interference, diffraction, dispersion, and the available resolving power.
Wavelength is one of the properties of a wave that is often easiest to measure. If a wave is standing still or not moving very quickly, then one may be able to measure the wavelength directly. For example, if you setup a standing wave on a string, then you can actually use a ruler to measure the wavelength. Often we measure the wavelength indirectly using interference patterns, especially when the wavelength is very small.
A sort of wavelength that scientists are often concerned with measuring is the wavelength of light, which is an electromagnetic wave. The wavelength of visible light is quite small, ranging from about 400-700 nm, but it can be measured through diffraction and interference. By measuring the wavelength of light we can determine the energy of the process that emitted that light. This allows us to determine the temperature of distant objects and the internal structure of atoms and molecules, in a process known as spectroscopy. The wavelength is important in most of the processes of physical optics.
In quantum mechanics, all particles travel as waves. One of the early steps in quantum mechanics was the discovery of de Broglie's relation, which relates the wavelength of a particle's wavefunction to its momentum.
1 One may assign a wavelength to any periodic wave; however, if it is not perfectly sinusoidal, then one may actually assign a number of different wavelength components to it through fourier analysis. The wavelength of periodic repetition would be the stongest component.
The concept of wavelength is something I take for granted so much I wasn't entirely sure what to put. Suggestions for improvements are more than welcome.