Instead of trying to say what flatness in analytic geometry means I'll give you some street-fighting tricks for recognizing whether a morphism of analytic spaces ( not necessarily reduced) $f:X\to Y $  is , or has a chance to be,  flat.

        
a) A flat map is always open.For example the embedding 
$\lbrace 0\rbrace  \hookrightarrow \mathbb C$ is not flat.More generally, the embedding of a closed (and not open !) subspace $X \hookrightarrow Y$ is never flat.

    
b) An open map need not be flat: think of the open  map $ Spec(\mathbb C) \to Spec(\mathbb C[\epsilon ]) $ from the reduced  point to the double point, which is open but not flat.  
 [It is not flat because the   $\mathbb C[\epsilon]$- algebra $\mathbb C=\mathbb C[\epsilon]/(\epsilon)$ is not flat : recall that a quotient ring $A/I$ can only be flat over $A$ if $I=I^2$ and here
 $I=(\epsilon) \neq I^2=(\epsilon)^2=(0)$]

An example with both spaces reduced is the normalization [see  g) below]  $f:X^{nor} \to X $ of the cusp $X\subset \mathbb C^2$ given by the equation $y^2=x^3$ .That normalization is a homeomorphism and so certainly open, but it is not flat : this results either from g) or from h) below. 
 
c) Given the morphism $f:X\to Y $ , consider  the following property:     
  $\forall x \in X, \quad dim_x(X)=dim_{f(x)} (Y)  +dim_x(f^{-1}(f(x)))$   $\quad (DIM) $      
We then have: $ f \; \text {flat} \Rightarrow f \;\text { satisfies } (DIM)$  
For example a (non-trivial) blowup is *not* flat.


d) For a morphism $f:X\to Y $ between connected holomorphic *manifolds* we have:

 $$f \text {  is flat} \iff f \text { is open} \quad \iff (DIM) \;\text {holds}$$ 
For example a submersion is flat, since it is open.

e) Given two complex spaces $X,Y$ the projection $X\times Y\to X$ is flat ( Not trivial: recall that open doesn't imply flat!). 

f) flatness is preserved by base change.

g) The normalization $f:X^{nor} \to X $ of a non-normal space is never flat.  
For example if $X\subset \mathbb C^2$ is the cusp $y^2=x^3$, the normalization morphism  $\mathbb C\to X: t\mapsto (t^2,t^3)$ is not flat.

h) Given a finite morphism $f:X\to Y $, each $y\in Y$ has a fiber $X(y) \subset X$ and for $x\in X$ we can define $\mu (x)=  dim_{\mathbb C} (\mathbb C \otimes_{\mathcal O_{Y,y}} \mathcal O_{X,x})$ . Now for $ y\in Y$ we put  $\; \nu (y)= \Sigma_{x\in X(y)} \mu(x)$ and we obtain :

$$f \; \text{ flat} \iff \nu :Y\to \mathbb N \text { locally constant}$$

[Of course for connected $Y$, locally constant = constant]

**Bibliography:**    
 A. Douady, *Flatness and Privilege*, L'Enseignement Mathématique, Vol.14 (1968)   
 G.Fischer, *Complex Analytic Geometry*, Springer LNM 538, 1976