What does it mean for a function to be holomorphic?

In fact holomorphic functions on a domain $U$ are identical to analytic ones, but the definitions shouldn't look that similar. If you've seen a similar definition of "analytic," it was cheating: an analytic function $f$ on $U$ is just one with a power series, $f(z)=\sum a_i(z-a)^i$ for some $a\in U$. It's an important theorem that this is actually equivalent to complex differentiability on $U$, which is very far from being the case over $\mathbb{R}$.

The "simpler terms" you request may be covered by recalling the definition of derivative: a function is holomorphic at $z_0$ if $\lim\frac{f(z)-f(z_0)}{z-z_0}$ exists as $z\to z_0$. If you write out what this means in terms of $f$ as a function on $\mathbb{R}^2$, you'll see both that $f$ is differentiable in the real sense and also satisfies the Cauchy-Riemann equations, which gives a third way to think about a holomorphic or analytic function.

A fourth way that may be more accessible to intuition is the amplitwist concept from Visual Complex Analysis by Tristan Needham. The fundamental insight is simple and compelling: the derivative of a complex function at a point is just a complex number, so that holomorphic functions must act infinitesimally by rescaling and rotating, because that's all multiplication by a complex number does. In particular, holomorphic functions are conformal: they preserve angles between curves (this is a fifth, partially independent, way to think about complex differentiable functions.) I highly recommend Needham's book as you look for more insights into this subject.