I am strongly agree with the answers of Prof.Brian G Higgins, Prof. Giuseppe Vairo, and Prof.Ahmad Hussain.

Check out Wikipedia for some examples:

http://en.wikipedia.org/wiki/Laplace's_equation

http://en.wikipedia.org/wiki/Poisson's_equation

For the numerical solution of incompressible Newtonian flows you can reformulate the Navier-Stokes equations so that the system of equations can be reduced to a pressure poisson equation ( the right hand side depend on the  velocity field). Thus given a velocity field and appropriate boundary conditions for the pressure field one can determine the pressure field. This is used in interative solvers. For a python implementation see : http://nbviewer.ipython.org/github/barbagroup/CFDPython/blob/master/lessons/16_Step_12.ipynb

In the field of computational analysis, the solution is obtained using iterative schemes, which indirectly solves Naiver stokes equation. Laplace, Poisson, Euler equations are all reduced form of Naiver stokes equation. Thus are significant for Mechanical engineers. For example in the area of computational fluid dynamics, these equations are widely used to obtain better and computationally faster results. You might be familiar about the various initialization schemes available in commercial CFD software, remember a better initialization yields faster results. Standard initialization just inputs the user defined values at all the nodes, this might not be useful every time. There is another kind of initialization called FMG initialization where a coarser grid is considered instead of actual grid and the Euler equation is solved, giving better initial guesses. And in hybrid initialization Laplace equations are solved for still better initial guesses.

This is one area where these reduced equations are used. 

Also analysis these equations and finding a way to solve them, finally gives better insight to solve the much complicated Naiver stokes equation with various boundary conditions for various solution domains .

Hope this helps you.

Regards.

Poisson and Laplace equations can be used to formulate the elastostatic problem of a rod under a torsion load. In detail, you obtain a Laplace governing equation with a Neumann boundary condition for the warping function (when you face the elastostatic problem via a displacement approach). Otherwise, via a stress formulation, you obtain a Poisson governing equation with a Dirchlet boundary condition for the stress function.

Poisson and Laplace equations are second-order elliptic equations that can model relevant problems like groundwater contaminant transport and petroleum simulation. In such problems the fluid flows in porous media are governed by these equations. There are several numerical methods for solving accurately and efficiently the problems.  

Engineering analysis involves the application of scientific analytic principles and processes to reveal the properties and state of the system, device or mechanism under study. Engineering analysis is decompositional, it proceeds by separating the engineering design into the mechanisms of operation or failure, analysing or estimating each component of the operation or failure mechanism in isolation, and re-combining the components according to basic physical principles and natural laws. Poisson and Laplace equations are partial differential equations with broad utility in electrostatics, mechanical engineering and theoretical physics.

You may find that Laplace equations are used to solve certain problems in the field of Controls and Vibrations etc.

Poisson equations are used solve problems in Electronics fields. 

It is used in Heat conduction equation, internal heat generation, diffusion process and many others related to Mechanical Engineering

I am strongly agree with the answers of Prof.Brian G Higgins, Prof. Giuseppe Vairo, and Prof.Ahmad Hussain.

The Laplace and Poisson equations are very significant  in Mechanical Engineering; just to mention few examples: see the following article entitled: "Local Fractional Poisson and Laplace Equations with Applications to Electrostatics in Fractal Domain" by Yang-Yang Li, Yang Zhao, Gong-Nan Xie, Dumitru Baleanu, Xiao-Jun Yang, and Kai Zhao  at http://www.hindawi.com/journals/amp/2014/590574/

Abstract:From the local fractional calculus viewpoint, Poisson and Laplace equations were presented in this paper. Their applications to the electrostatics in fractal media are discussed and their local forms in the Cantor-type cylindrical coordinates are also obtained.

Another example is: Poisson's Equation in Electrostatics

www.nhcue.edu.tw/~jinnliu/proj/Device/3DPoisson.pdf

Behind the significance and applications of the Laplace and Poisson equations mentioned by my predecessors, I would to mention the application in the area of mathematical modelling of different systems in structural mechanics, flow in porous media,..., etc!

An example of application is given by following paper:"A Noniterative Numerical Solution of Poisson’s and Laplace’s Equations With Applications to Slow Viscous Flow"!

http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=1432909

in potential flow analysis, which is very significant to mechanical engineering, both stream function and velocity  potential function are obeying the Laplace equation.

Solution for Airy's stress function in plane stress problems is a combination of general solutions of Laplace equation and the corresponding Poisson's equation. They are useful in the solution of bi-harmonic operator in Kirchhoff's theory of bending of isotropic plates.  

Laplace equations are used for the description of steady states.