The main tool of numerical analysis is a computer. A central issue is how numbers like pi or the square root of 3, whose exact values typically require infinite quantities of digits, are represented in the computer. It is not possible to be exact since there is not enough memory to represent infinite digits. Therefore, numbers are chopped after a certain number of digits or rounded. Numerical analysis studies how to control the errors that can be propagated and amplified by either method.
Infinite processes must be approximated in the real world by finite ones. Think of the problem of representing the square root of 2. If accuracy is not an issue, 1 might do. But 1.4 will be better. Better still is 1.41 and yet more accurate is 1.414. We might continue the process indefinitely, each time getting closer to the real value of the square root of 2. But there isn't time enough nor memory in all the computers in the universe to get it exactly. So we stop somewhere and decide that some representation, say, 1.414213562373 is close enough to the real value to be what we mean by the square root of 2. Of course, that means there will be some error inherent in our calculations. The error resulting from stopping an infinite process at some point is called "truncation error" and controlling and understanding it is a central issue in numerical analysis.
One of the main applications of numerical analysis is approximating roots of polynomials when exact solutions can't be found. This is usually done by iteration--that is by choosing a first approximation (often an educated guess) and applying a formula. The resulting value is put through the same formula and the process continues until the value attained after the most recent iteration is within a preset tolerance of the exact value. One important iterative process studied in numerical analysis for instance is Newton's method for approximating roots.
In calculus, you learn to calculate the area under curves in the plane by finding an important quantity called the "anti-derivative." The problems in your book are carefully chosen so that the anti-derivative can be found. But in the real world, this mysterious anti-derivative doesn't always exist or is impractical to find. In numerical analysis you learn to approximate the area under curves even when the anti-derivative doesn't exist. This is accomplished by using trapezoids or rectangles or even more general shapes to estimate portions of the area and then adding them up. The process of estimating such areas is an important topic in numerical analysis and is called numerical quadrature.
In numerical analysis, you study the the process of solving systems of linear equations in great detail and consider different strategies for dealing with roundoff errors that result in such problems from computer arithmetic. The number of steps required for elementary linear algebra techniques using matrices is calculated and compared to alternatives to determine which would be best to use. Another important topic is finding the best line that fits a collection of data, also called the least squares problem.
In upper level or graduate mathematics courses, students spend considerable time approximating solutions to differential equations. These are equations involving first and higher order derivatives and their powers. A derivative is the rate of change of one variable with respect to another. An example of a derivative is the speed of a car and the rate of change of its position with respect to time. Differential equations can be extremely hard to solve and may not lend themselves to exact solutions. But numerical analysis gives a means of approximating a solution to any degree of accuracy.