What is FEA?

Finite Element Analysis (FEA) is a numerical method used to predict the response of structures to various environmental loads.  The basic idea of FEA is to take a continuous system and to discretize it. 
FEA begins with a computer model of the given geometry.  The geometry, which is continuous, is then subdivided into small pieces called elements.  Elements are simple shapes (such as triangles, squares or cubes) that are connected to one another using nodes.  The collection of elements and nodes is referred to as a mesh.
Below is a simple representation of nodes and elements:
Using nodes and elements, complex systems can be represented so that a FEA can be completed:

Using elements, complex partial differential equations that describe the behavior of the structures can be reduced to a set of linear equations that can easily be solved using the standard techniques of matrix algebra.  Material information, structural information and loading conditions are also applied to the model and the response is predicted.