bifurcation. what does it all mean?

Ihave been looking at the bifurcation diagrams. you know, the one that
have this line that seems to split and then split again then turns
into weird mesh of dots..

what does this mean? 

how do they make these diagrams? 

I guess bifurcation occurs when you have abrupt changes in systems
with changes in paframeter.

so which is the parameter axis and the system axis?  or am I not
looking at the figures right?

baffled. 

thanks all 

john