Process: When animals cells go through cytokinesis it called Cleavage.

How does it work?

Cleavage is when the cell membrane is pinched inward at the equator, this forms a thing called 'Cleavage Furrow' 

As it deepens it splits the cell into two daughter cells!! This is able to happen by a structure called contractile ring ( This is made up of actin filaments and myosin motor proteins). (Michael Post, Cytokinesis: Plant vs. Animal Cells, 2004)

Process: When plants go through cytokinesis from a thing called Cell Plate.

How does it work?

A new cell wall grows in a inward motion from the center of the cell, this divided the new cell into two daughter cells.  

Golgi vesicles carry the cell wall material to the center of the cell. 

These vesicles are fused at the midline creating the structure called Cell Plate. (Michael Post, Cytokinesis: Plant vs. Animal Cells, 2004)