Financial engineers are generally hired by banks, insurance companies, and other asset managers (hedge funds, pensions, etc) to design math models that lower financial risks and make better returns.
They use modelling techniques like convexity, duration, value at risk, simulations, linear programming, and other processes to determine future returns, loss scenarios, etc.
Topics such as statistics, calculus, engineering, physics, and advanced math or "quant" abilities are the fields they usually study, along with finance, accounting, and economics.
Good computer skills are necessary (a ton of data and computing power is needed to run most of these models), but good math skills are a must.
They usually have a strong background in modern portfolio theory, option theory, risk management, etc, along with their math and science backgrounds. And yes, they tend to make a lot of money.
Wikipedia and Investopedia have a number of good introductory articles on financial engineering. Look up "computational finance," "quantitative finance," "Black Scholes Option Pricing," "Value at Risk" and "Risk Management," and "the Greeks," for more information.
Schools like MIT, the University of Chicago, and many of the Ivies have terrific programs in Financial Engineering, Quant Finance, and Quant Economics.