A Laplace transform is useful for turning (constant coefficient) ordinary differential equations into algebraic equations, and partial differential equations into ordinary differential equations (though I rarely see these daisy chained together).

Let's say that you have an ordinary DE of the form

$$a y''(t) + b y'(t) + c y(t) = f(t) \quad t \gt 0$$
$$y(0)=y_0$$
$$y'(0)=p_0$$

Then the above equation becomes

$$(a s^2+b s+c) \hat{y}(s) - [a y_0 s + (a p_0 + b y_0)] = \hat{f}(s)$$

where $\hat{y}$ and $\hat{f}$ are Laplace transforms of $y$ and $f$, respectively. Note that we have converted the ODE into an algebraic equation in which we solve for $\hat{y}(s)$. We find $y(t)$ by inverse Laplace transformation, which is usually accomplished through tables, or contour integration if there is facility with that. Note that the initial conditions are built right into the equation we solve.

There are numerous examples for using Laplace transforms in PDE's. Here is a case I did in which I used LT's to solve the heat equation in two dimensions.