Finite Element Analysis (FEA) is most associated with FEA software, which is generally treated as a black box that generates useful outputs when the necessary inputs are provided. However, a basic understanding of FEA theory can make this black box much easier to use. This chapter is a concise summary and description of the Finite Element Analysis theory. It will introduce and discuss the fundamental concepts of FEA theory to help in understanding the theory behind the software driving this black box. Finite Element Analysis is a numerical approximation technique for solving deformations of objects. Every object is divided into many small blocks, called elements. Each element has several nodes at the corners, edges, and outer surfaces of the element. Approximation deformation functions inside a finite element can be guessed and expressed in terms of nodal deformation. Scientific principles, such as the minimum potential energy principle, are used to minimize the total error in constructing element properties, which converts the set of partial differential equations into a set of linear algebraic equations. The shared nodes of elements are pinned together so that an assembly of many elements can be used to represent the original continuous objects. This chapter provides several examples of how to guess a deformation function inside an element by using the concept of a shape function. One example of a uniaxial stepped bar under axial loads is fully demonstrated and solved manually by using the FEA method. This example will significantly facilitate us to have a better understanding of the fundamentals of FEA theory.