Precisely to define applied mathematics is next to impossible. It cannot be done in terms of subject matter: the borderline between theory and application is highly subjective and shifts with time. Nor can it be done in terms of motivation: to study a mathematical problem for its own sake is surely not the exclusive privilege of pure mathematicians. Perhaps the best I can do ... is to describe applied mathematics as the bridge connecting pure mathematics with science and technology.
Applied mathematics is, by definition, interdisciplinary. See especially: Mathematics, Statistics and Probability, Physics, Engineering, Computer Science and Engineering, Psychology, Economics, Business, Finance, History and Philosophy of Science, Leisure Reading — Science and Technology.
Related topics include: Your Scholarly Identity (ORCID), Designing Documents with LaTeX, Research Data Services, Scholarly Communication, Open Access Publishing at Notre Dame.
The most accessible encyclopedia, with definitions, is Wolfram Math World. Another authoritative source, although very technical, is the Encyclopaedia of Mathematics, edited by M. Hazewinkel. Print volumes may be requested from the Hesburgh Libraries Annex. Another work related to applied mathematics is the Encyclopedia of Statistical Sciences.
Improving Your Proofs
There are a number of books on writing better mathematical proofs. Here are two that may help:
One of the best sources for biographies of mathematicians is the MacTutor History of Mathematics Archive.
Careers in applied mathematics are discussed in depth in the Bureau of Labor Statistics' Occupational Outlook Handbook's group of Math Occupations.