What does it mean by a complex function f(z) ?

In complex calculus, we define the complex number  z as,

z = (x+i y),  where x and y are real numbers and  i = sqrt(-1)

The plane formed by x & y  is called as the ' z-plane ' .  The function  f(z)  is again a complex number, as  z is complex and we define the function  f(z) as follows.

f(z) = w = u + i v , where u & v are real and depend upon the value of z. Hence they are functions of x & y.

i.e., u = u(x, y) , v = v(x, y) ;  and the plane formed by u and v is called as the w-plane. So, the function  f(z) can be viewed as a mapping from the z-plane to a w-plane, for a specified mapping rule (for  a defined function mathematically). 

Lets see some examples to make it clear how the function f(z) maps certain geometries in z-plane into a w-plane under certain specified mapping rule.

Ex. 1.  Say we've y = mx (a straight line passing through origin ) in the z-plane, lets' see how this geometry maps into a w-plane, under the mapping rule  f(z) = z^2.

we've f(z) = z^2 = (x + i y)^2 = (x^2 - y^2) + i (2xy)

that gives  u(x, y) = (x^2 - y^2)  and  v(x, y) = 2xy. 

But we've the relation y = mx  in z-plane, so, by substituting this relation into u & v,  we get,

u = (1-m^2)  x^2 ,  

v = 2m  x^2

by eliminating x from the above two eqns, we can obtain the relation b/w  u & v. as,

v = [2m / (1-m^2)] u , which is also a straight line with slope 2m / (1-m^2).

To represent it  pictorially,

Similarly, one can think of any arbitrary mapping rule and try to visualize how does it maps the given geometry in the z-plane into a w-plane. For example, if you take mapping rule to be exp(z),  then verify yourself that, it maps into a spiral in the w-plane.
If its not possible to obtain the relation directly  b/w  u and v  by the elimination of variable x, then one can go for either mathematica or any numerical technique to see the parametric plot  b/w  u & v.

That tells you the geometry in w-plane.