Harmonic analysis is a branch of mathematics that looks at the theoretical foundations of digital signal processing. A continous signal can be drawn as a wave, or as a combination of several waves. Fourier transforms and Fourier series are among the main tools used for signal analysis. Today, this field has many uses, which also include quantum mechanics and neuroscience.
In essence, harmonic analysis looks at locally compact groups. The Lebesgue measure is a way to assign a measurement to a subset of the Euclidean space. For real numbers there is the Haar measure, which can perform this task. This measure allows to use Fourier analysis to model the goups and their properties.
The term harmonics is also related to Eigenvalues, in the case where the frequency of one wave is an integer multiple of the frequency of another wave. This is the case of the harmonics that are used for musical notes. Later the term was generalized.