Where can I learn to define mathematical terms? For example, take the following:

The radian measure of a central angle of a circle is defined as the
  ratio of the length of the arc the angle subtends, s, divided by the
  radius of the circle, r.

This is one of the many definitions of a radian. 
Firstly, what is this kind of definition, where temporary, conventionally-named variables are introduced such as the "r" in "radius of the circle, r", known as?
And is the skill of writing such definitions developed intrinsically? Or are there formal guidelines, a 'nomenclature' of some sort, where the defining-process is formally described?
 A: You need to recognize what you are trying to communicate, and what your reader possibly doesn't know, and then you need to learn how to be a clear writer. All of these pieces require experience.
There is a formalism in making sure that you have given a definition that is accurate to what you want, but this is more a question of how to write clearly and convey information. 
Indeed, in heavily formalized mathematics, we simply "skip" definitions - the languages of first- and second-order mathematics does not contain a provision for adding definitions, and everything can be done without definitions. Definitions themselves are just shorthand for our convenience. They are a means of human communication.
The above definition is an interesting example, because it requires a separate step to get the result we all "know," that the radians measurement is independent of the radius of the circle. You can't actually write the definition without that problem. No matter what definition you use, you will need to prove something. That's because the independence of the radius choice is a feature of Euclidean geometry, not inherent in the definition alone.
