Natural units are what groups of scientists in a common field of research use to make maths problems easier for themselves, and their work less intelligible to others.
Natural units result from taking a certain physical property or measurement and assigning it a value of 1. An example is the Astronomical Unit: the distance1 from the Earth to the Sun is 1 AU. This makes a lot of calculations very easy for some people; it's also an international effort to make conversion difficult again for everyone else. The problem is that you have to regularly update every physical constant to match the new units. Which means that instead of getting any real work done, most scientists just sit around all day converting constants into special units and back again.
Students are usually left in total confusion as to which unit or constant to use in practice, because most professors believe that it is just as important to learn arbitrary notation as it is to understand the topic at hand.
Please, stick to metres, kilograms, and seconds. Unless presented with the direst need, or you really want to.