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Correct me if this question is too broad:

I heard that real harmonic functions are the analogon to holomorphic functions. I know that they have a lot of properties in common (e.g. analyticity) but what is their connection exactly?

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    The real and imaginary parts of a holomorphic function are each harmonic. – 2017-01-31

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Let $f(x+iy)=u(x,y)+iv(x,y)$ be a holomorphic function. Then you can calculate, by using the Cauchy-Riemann equations, Schwarz Theorem and that f' is holomorphic, that $Re(f(z))=u(x,y)$ is harmonic, in other words $\Delta u(x,y)=0$. Same goes for the imaginary part $Im(f(z))=v(x,y)$ i.e. $\Delta v(x,y)=0$.
The other way around: If you have a simply connected domain (think it is simply connected, youd have to look that one up), every harmonic function $u(x,y)$ is the real part of a holomorphic function. Same applies for the imaginary part.