The cubic Mandelbrot set comes from iterating z = z^{3}+c. The basic structures are the same as in the regular Mandelbrot set but everything has 3-fold symmetry.
Calculating images of the cubic Mandelbrot set takes significantly more work than the quadratic version because more multiplications and additions are involved. For that reason, not many animations have been made with it.
Two deep-zoom animations have been published on this site so far:
The cubic Mandelbrot set resembles, but it definitely different from, the tricorn ("Mandelbar") fractal. Both have three-fold symmetry at low magnifications, but the tricorn fractal is fundamentally a second-order structure, and it contains mini-brots that are exactly the same as the usual second-order mini-brots, while the cubic Mandelbrot set only has third-order mini-brots.
Note on size and magnification: The sizes here (and on the Animations page) are the actual size of the smallest dimension of the image (usually vertically) in the complex number plane. Some programs describe image sizes by "magnification" which is usually related to the reciprocal of the image size. A size of, say, 1E-100 corresponds to a magnification of 1E+100. Some software uses the half-height of the image, so there may be an additional factor of two involved in conversion.