Assuming you are asking for laplace equation : Given a complex 3D shape, with uniform resistivity, to find total resistance across two surfaces of the shape - the most accurate method / algorithm to extract the resistance is to solve Laplace's equation with given boundary condition numerically.
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If your question is on Laplace Transform : LT is an integral transformation of a time domain function f(t), into a complex frequency domain. So, when a known signal f(t) passes through a Filter [with known frequency response characteristics ] - it is much easy [for human mind] to transform signal into into frequency domain - and for final result, after passing through the Filter, get the reverse transform done to get back signal form in time domain.
If your question is on Laplace Transform : LT is an integral transformation of a time domain function f(t), into a complex frequency domain. So, when a known signal f(t) passes through a Filter [with known frequency response characteristics ] - it is much easy [for human mind] to transform signal into into frequency domain - and for final result, after passing through the Filter, get the reverse transform done to get back signal form in time domain.
There are many types of integral transformations. Standard terminology is
Fourier transform: Time domain <=> Frequency domain
Laplace Transform: Time domain <=> Laplace domain, s-domain or also sometimes frequency domain.
The whole point of the transformations is that various operations become easier. The most typical may be that convolution in the time domain becomes multiplication in the s-domain. The working of filters can be sometimes understood more easily by looking at the transforms of your signals.
To put it more simply: integral transforms can change integral/differential equations into algebraic equations which are usually simpler by transforming differentiation and integration into multiplication and division (just like logarithms turning multiplication/division into addition/subtraction).