There are two main catagories of cells:  prokaryotic and eukaryotic.  Prokaryotic cells are more primitive and do not have a nucleus.  
Eukaryotic cells are more advanced and do have a nucleus and its that nucleus which is the basis of all animal and plant life.  The nucleus is one of the organelles inside the cell.  The nucleus contains chemical codes, DNA , which direct the activities of the cell and tell it what kind of proteins to make.  The DNA is also the means by which cells are able to reproduce themselves in an exact way.

DNA molecule

DNA resides in packets, called chromosomes , inside the nucleus of the cell.  Every cell has a certain number of chromosomes inside its nucleus.  This number is different for each animal species.  In humans, the number is 46.  What does DNA do?  First of all, DNA is a set of instructions for making protiens.  Proteins are the building materials of life and are themselves made of organic chemicals called amino acids.  Another vital function of DNA is its ability to reproduce itself exactly so that a species can continue from one generation to the next.  DNA has two parts, part from thye mother of the organism and part from the father.  When it reproduces, DNA uses information from both these parts.  Sometimes the code contained in DNA is altered by something external such as radition from a cosmic gamma ray.  This causes what is called a mutation and is responsible for the process of evolution.

All animal organisms start from just two cells, a sperm cell and an egg cell.  In order for a complete animal to develop, the cells must increase their numbers by reproducing themselves.  The cells do this via a process called mitosis.