Generating random numbers from computers is essential to the development of certain kinds of software. Anything from modelling the environment, to a lottery machine, to determining the value of loot in a chest in an RPG, will require random number generation. At first it may seem strange that computers, which are capable of producing massive amounts of digits in a short time, would not be able to produce random numbers. The difficulty is that the computers we use, are constructed specifically to follow logical steps deterministically, so to generate numbers that are truly random from a system like our computers is virtually impossible. True randomness cannot be obtained using arithmetic operations, which is exactly what our computers perform. When some sequence of numbers is random then it is not possible to predict what the next digit will be, and since computers use a set of logical steps to create any new number it is theoretically possible to predict it. Nonetheless programmers take advantage of features for creating random numbers all the time, the most widely used programming languages, provide libraries that can generate 'random' values. Any role-playing game for example, that you might use, needs random values to determine the number of gold you'll find from some locations, what items dead enemies will drop, etc. So how is it possible that even though the very nature of computers makes it impossible to exhibit random behaviour, that many programs include simulations of randomness?
The answer is pseudo-random number generators(PRNG), these are programs that follow a certain algorithm to simulate random behaviour. The numbers they generate, cannot be by definition truly random, however for many uses it is a good enough approximation. The libraries in many of todays programming languages that generate random digits use PRNG's. The basic idea is to perform some arithmetic operations on a number in sequence, to make it seem like it's random. A PRNG always starts with an initial value called a seed. It will then perform operations on the seed to make the final number different from the original value. So when the PRNG is supplied with seeds that are in sequence, it will return numbers that seem random, because the operations that are performed on each number should change it to make it so the returned numbers are no longer following any pattern. PRNG's are also periodical, they have a certain amount of numbers they produce, and after that the sequence starts to repeat itself. It is usually not a problem to make the period very large, so that the application that relies on it will never start to get repeated sequences.
A very well known PRNG is the Mersenne Twister(1), it is the default PRNG for Python, Ruby, PHP and many others. It has a period of 2^(19937−1), which is a Mersenne prime number, hence the name of the generator. Invented in 1997 it was largely superior to PRNG's like C's rand or Java's Random. Furthermore it passes certain statistical tests for randomness so it is a very reliable generator as it is very successful at simulating genuine randomness.
These programs have been used extensively and are relied upon by many applications, it is important to note that when truly random behaviour is needed then PRNG's are not sufficient. John von Neumann famously said: “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." In this situation he was referring to depending on PRNG's or other arithmetic approaches to create truly random digits, which is not a good idea. Rather it is possible to use true random number generators, that are based on the idea of extracting physical phenomena that are believed to be random and using them to generate random numbers. For example atmospheric noise, or radioactive decay, etc. Even though it is possible to extract truly random numbers from the outside world, most applications that use random numbers use PRNG's. It is because of their convenience, there is no need for any extra devices, or input, all it needs is a sequence of numbers that it later transforms into a pseudo-random sequence.
1: Link to the implementation of Mersenne Twister. This is the latest version from the original creators: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/MT2002/CODES/mt19937ar.c