about laplace transforum

The s domain characterise the transfer fuction of a system
The Fourier transform is a special case of the Laplace ; down the j-omega axis (reel =0)

Fourior transform and Laplace transform are used for solving differential and integral equations in the engineering. Laplace transform, transforms the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. It is always easy to solve a problem in frequency domain as compared to time domain.