Well, LU is one of the fastest ways computers use to solve linear systems of equations of the form $\mathbf{Ax}=\mathbf{b}$. It's especially useful if you have a lot of RHS's for which you want to find solutions, because once you find L and U, you're off to the races (they don't change for different RHS's).
One good application would be in solving a large circuit. You could easily get 10 or 20 linear equations to solve, and the computer can do that faster than a person, typically.
Solving linear systems is important in modeling parts, like a landing gear for an aircraft, for example. There you'd typically do a Finite Element Analysis, resulting in a large, sparse linear system to solve. For those size problems, you're not going to use an exact method like LU, but methods tailored to sparse matrices.