Numerical analysis is making use of equations to calculate quantities, not by solving for those values so as to carry out the equation's specified arithmetic, but by devising a way to get closer and closer to the answer by repeating some arithmetic. Essentially, you devise a different set of equations that are solvable, and which through repeated use, bring you closer to the original equation's answer. This can be useful or vital if algebraically solving the original equation is difficult or impossible.
Calculating a square root offers a straight-forward example: guess an answer, square it, and adjust your guess up or down as the resulting square indicates. Systematic methods of adjustment can be devised, e.g., if one trial was too big and another too small, use their mean as your next guess. Hitting an exact answer may require luck, but mere persistence gets you as close as you need.
Another numerical analysis technique, termed numerical integration, is of use if you have no straight-forward formula for some function but do have such a formula for the function's slope, and have at least one point, i.e., a number and the functional value associated with that number. Given that value and the calculable slope, the strategy consists of using them to estimate the value of a nearby point (the functional value of a number near the original one), then repeating the process to map out values of the function over an interval of interest. The slope is calculated repeatedly throughout this interval, each time a small step further, and the smaller the steps (i.e., the more calculation), the more accurate the estimates (or rather, the procedure is only practical if the function has this property).
Numerical analysis can require orders-of-magnitude more arithmetic than more straight-forward methods: a tiny change in a solvable equation that makes it not so, might easily make a calculation require a million times more arithmetic. Currently it is natural and common to use computers to do this, with programs that devise appropriate new guesses. Often the term computation is used to mean doing numerical analysis with a computer (because it requires so much computation) and in fact, numerical analysis was a key motivator in the development of the computer as we know it. The methods of numerical analysis are referred to as numerical methods, and their development is an entire science of much interest, because no matter how much computing capacity is available, more efficient and well-suited methods can solve more problems with that capacity. Models using these methods are termed numerical models.
In contrast to numerical analysis, using equations by manipulating them to solve them algebraically for the quantities you need is called solving analytically, or using analytical methods. With the growth of computer capacity, numerical methods are often used even if a problem could be solved analytically. A third method of tackling math problems, which numerical analysis has displaced to a degree, is analog computers, such as slide rules.
The use of numerical analysis predates computers, but a much more limited set of problems could be tackled in such a manner, even using weeks or years of people doing the arithmetic. Numerical analysis was often used for deriving generally useful answers, such as digits of irrational numbers like π and e, values of trigonometric functions for many possible inputs, and of roots, the answers commonly published as tables.
Some types of methods that have been developed: