This is a longer question.
In the physical sciences
In the 19th century scientists used the idea of random motions of molecules in the development of statistical mechanics in order to explain phenomena in thermodynamics and the properties of gases.
According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random. That is, in an experiment where all causally relevant parameters are controlled, there will still be some aspects of the outcome which vary randomly. An example of such an experiment is placing a single unstable atom in a controlled environment; it cannot be predicted how long it will take for the atom to decay; only the probability of decay within a given time can be calculated. Thus quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories attempt to escape the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, unobservable (hidden) properties with a certain statistical distribution are somehow at work, behind the scenes, determining the outcome in each case.
 In biology
The theory of evolution ascribes the observed diversity of life to random genetic mutations some of which are retained in the gene pool due to the improved chance for survival and reproduction that those mutated genes confer on individuals who possess them.
The characteristics of an organism arise to some extent deterministically (e.g., under the influence of genes and the environment) and to some extent randomly. For example, genes and exposure to light only control the density of freckles that appear on a person's skin; whereas the exact location of individual freckles appears to be random.
Note that this effect isn't limited to physical characteristics. Sexual orientation also appears to have a random element, for example. In identical twin studies, such twins are more likely to have the same sexual orientation than two randomly chosen individuals in any given population. This correlation is attributable to genetics and chemical influences within the womb if the twins are adopted and raised in separate environments, but could be due to either genetic or environmental factors if they are raised in the same environment. However, even identical twins raised in the same environment do not always have the same sexual orientation. In cases where there is a difference in sexual orientation between the two, this is typically ascribed to a random element, although this could also result from a pattern of events more complex than is currently understood.
 In mathematics
The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling but soon in connection with situations of interest in physics. Statistics is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation it is necessary to have a large supply of random numbers, or means to generate them on demand.
Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits is random if and only if it is shorter than any computer program that can produce that string (Kolmogorov randomness) - this basically means that random strings are those that cannot be compressed. Pioneers of this field include Andrey Kolmogorov, Ray Solomonoff, Gregory Chaitin, Anders Martin-Löf, and others.
 In information science
In information science irrelevant or meaningless data is considered to be noise. Noise consists of a large number of transient disturbances with a statistically randomized time distribution.
In communication theory, randomness in a signal is called noise and is opposed to that component of its variation that is causally attributable to the source, the signal.
 In finance
The random walk hypothesis considers that asset prices in an organized market evolve at random. Other so called random factors intervene in trends and patterns to do with Supply and Demand distributions. As well as this, the random factor of the environment itself results in fluctuations in stock and broker markets.