From a theoretical point of view, Laplace transform is not that different from Fourier transforms; all you need to do is to go from the real axis to the imaginary axis on the complex plane. For many practical problems, the Laplace transform is "easier" because the integral defining the Laplace transform converges pretty fast.
One application of Laplace transforms is for solving ordinary differential equations with constant coefficients. Here is the basic procedure. Under the Laplace transform, such a differential equation "becomes" an linear algebraic equation in the Laplace transform. One solves the algebraic equation to get the Laplace transform of the solution of the ordinary differential equation, then performs an inverse Laplace transform to find the solution.