Given a point z with spherical coordinates (phi,theta,rho) define a new point S(z) has the spherical coordinates (8 phi,8 theta,rho^{8}). Now define T(z) = S(z) + c, where c=(u,v,w) is a parameter. The Mandelbulb is the set of parameters (u,v,w), where a successive application of the map T to (0,0,0) produces a bounded orbit.
This is analogue to the Mandelbrot set which is T(z) = S(z) + c, where c=(u,v) is a parameter and S(z) maps a point with polar coordinates (theta,r) to (2theta,r^{2}).
On Youtube you can watch wonderful movies of the Mandelbulb set. Here are three 1, 2, 3 and displayed here:


