model reality

Modelling is the key concept to make things in computer graphics look real, or almost real. To make an application 'feel' real you have to model reality as good as possible. It depends on your application what to model and how exact you want to model reality.
As computers are very fast in computing and comparing, mathematics is the best choice to map reality. But computers have one big disadvantage, a limited memory. Thus the computer has its limits and you are limited in the way how to model the world. Numbers represented in computations may not be correct. Some numbers can be mapped exactly others not. We have to design our models carefully when we compute with limited numbers. This design leads to either a useful or useless model.
Simple but stable models will often lead to a more realistic feeling than models that compute everything but are slow and unstable. There is a tradeoff between a good model and interaction. Nowadays computer hardware is fast enough to make things easier and so the models become more complex and realistic.

design issues

Keep always in mind that during computation round-off errors and wrong interation steps will lead to wrong numbers which directly leads to an useless model. I will point out such problems in each subsection and give some hints how to avoid them. Each type of modelling will have its own advantages and disadvantages, so you have to decide which methods fits the best for your needs.
I will not go into detail. Wherever possible I will provide you links or keywords to search for which cover the given problem.