1. Introduction
The purpose of Laplace transformation is to solve different differential equations. There are a number of methods to solve homogeneous and non-homogeneous equations but Laplace and Fourier transform are used widely. Laplace transform comes in to use when we have to solve the equations that cannot be solved by any of the previous methods invented.
It can be used to solve various algebraic problems. Laplace transform is more expedient when it comes to non-homogeneous equations. It is one of the easiest methods to solve complicated non-homogeneous equations.
Definition
The Laplace transform converts integral and differential equations into algebraic equations. Although it is a different and beneficial alternative of variations of parameters and undetermined coefficients, the transform is most advantageous for input terms that piecewise, periodic or pulsive.

2. In simple words, Laplace transform converts time domain signal into frequency domain, which is depicted more clearly in below figure: