Numerical analysis is the study of how to approximate continuous functions using rational numbers. The study of algorithms is an essential aspect of numerical analysis, because the result of computations may depend on the order in which the computations are performed. With the advent of high-speed digital computers, numerical analysis has become an important part of applied mathematics.
It predates computers by many centuries. Taylor approximation is a product of the seventeenth and eighteenth centuries that is still very important. The logarithms of the sixteenth century are no longer vital to numerical analysis, but the associated and even prehistoric notion of interpolation continues to solve problems for us.
The effect of round-off error is partly quantified in the condition number of an operator. Subtraction of two nearly equal numbers is an ill-conditioned operation, producing catastrophic loss of significance. Using well-conditioned operations helps achieve numerical stability.
An important part of Numerical Analysis is concerned with computing (in an approximate way) the solution of Partial Differential Equations. This is done by first discretizing the equation, bringing it into a finite dimensional subspace, then solving the linear system in this finite dimensional space. The first stage is done by the Finite Element method, finite difference methods, or (particularly in engineering) the method of Finite Volumes.
The Finite element method is the more powerful approach to numerical differential equations, but mathematicians prefer finite difference because the theorems are easier to prove. Shame on them. This is hardly true. Finite elements is based on a vast number of results in functional analysis. FD in contrast is very messy, involving chasing around terms in Taylor series. But I am going to fix this momentarily :)