Harmonic functions appear most naturally in complex analysis and extends the concept of analytic functions.
The Cauchy-Riemann equation together with the conjugated Cauchy-Riemann equation shows that the sum of an analytic function and an anti-analytic function is harmonic and in fact every complex harmonic function can be written as such. In particular the real/imaginary part of an analytic function is harmonic.
Harmonic functions satisfy the Liouville's theorem and maximum principle, in any dimension.
It should be mentioned that harmonic functions can be generalized one step further to the class of sub-harmonic functions which satisfy $$\Delta f\geq0$$ which also satisfy the maximum principle.