In his theory (from 1859 until his death), Darwin proposed two evolutionary drivers: Natural Selection and Sexual Selection to explain how the variety of life arose from "common descent".  Under the theory of Natural Selection, the environment preserves individual organisms on the basis of their carrying traits - which confer survival advantages on them. Conversely, the environment eliminates individual organisms on the basis of their carrying traits - which confer survival disadvantages on them.
"As many more individuals of each species are born than can possibly survive; and as, consequently, there is a frequently recurring struggle for existence, it follows that any being, if it vary however slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have a better chance of surviving, and thus be naturally selected." -- Darwin
The theory of Sexual Selection runs: (1) in some species, males compete for the access to females. (2) in other species, females select attractive reproductive traits, displayed by males. These "elablorate male traits" (also called "ornaments") convey reproductive advantages on males, yet they also convey survival disadvantages on them. (Natural Selection cannot explain the appearance and persistence of "male ornaments" - as well as other features of organisms, but sexual selection can.) Mating between the selecting and the selected individuals leads to non-random, differential reproductive outcomes, resulting in genetic novelty and in more raw material on which Natural Selection operates. 
"The sexual struggle is of two kinds: in the one it is between the individuals of the same sex, generally the males, in order to drive away or kill their rivals, the females remaining passive; while in the other, the struggle is likewise between the individuals of the same sex, in order to excite or charm those of the opposite sex, generally the females, which no longer remain passive, but select the more agreeable partners." -- Darwin
Darwin predicted that - since his theory lacked "a mechanism of inheritance", it would fail should a new theory come-along, identifying one. After the simultaneous re-discovery of Mendel's laws in 1901 by Carl Correns, Erich Tschermak and Hugo de Vries, much of Darwin's theory was abandoned (or more accurately backgrounded) for a brief period of time. Most geneticists of that day concluded that Mendel's laws - as applied to evolution (in a theory dubbed "mutationism") - cashed-out the fossil record, speciation and diversity better than Darwin's did. Many geneticists at the time explicitly rejected Natural Selection. In short order - however, mutationism failed more rapidly than Darwin's did.
In 1918, Sir Ronald Aylmer Fisher (against the scientific consensus) proposed that Darwin's biology was compatible with Mendel's laws and that any viable theory of evolution required both. Fisher's hypothesis turned-on the pivot that Mendelian inheritance, via discrete alleles, "sums up" into gradual and continuous changes in populations over short periods of time (and such changes accumulate into speciation events over geological periods of time). These Darwinian outcomes, arising from Mendel's laws and Darwin's drivers, were mathematically predictable (and mappable) with the statistical techniques, invented by Karl Pearson and himself. The resulting theory of Population Genetics dealt a decisive blow to the Mendelian dominance held over biology in the early 20th century.
Under the Great Synthesis from the 1920s to the 1950s, Darwinian biology and Mendelian genetics were formally fused by a blizzard of papers and books into what is known as "the Modern theory of evolution". However - even Darwinian biology and Mendelian genetics conjoined do not adequately explain the variation and diversity shown by living things. Theodosius Dobzhansky pointed this out - clearly and early on - due to his experiences in studying fruit flies on many continents. He knew that the emerging synthetic theory did not account for the bewildering number of fruit fly species he encountered. Dobzhansky proposed "isolating mechanisms" (finely-tuned, selectable sex traits and other factors which disincline populations from inter-breeding), and - initially - he tapped genetic drift to plaster over the cracks between the deficiencies in the new theory and the measure of species-diversity detected in nature.
In 1929, the process of genetic drift was identified by Sewall Wright. For decades, it was controversial and downplayed (especially by Fisher). Only in small populations was genetic drift thought to contribute significantly to evolution. In the late 1960s, drift vaulted back into prominence - again - when a series of important experiments was conducted. The results of these experiments suggest that the riddle of the "diversity deficit" between Modern evolutionary theory and what is observed in nature may have been solved. One conclusion from the experiments was that "some if not most of the changes in the genetic material are caused by [random] genetic drift." - not by the non-random drivers of Natural and Sexual Selection. 
To be clear: Without the driver of Natural Selection, evolutionary theory fails to explain the existence of "utilitarian" (adaptive) traits - which organisms carry. Without the driver of Sexual Selection, evolutionary theory fails to explain the persistence of "non-utiltarian" (disadvantageous to survival) traits - which (primarily) male organisms show. Without Mendel's laws, evolutionary theory fails to explain heredity. And - without the driver of genetic drift, evolutionary theory fails to explain the "neutral" (non-adaptive) traits, invisible to selection - which organisms carry.
Barely sketched above (Darwinian biology + Mendelian laws of inheritance + genetic drift) is a very powerfully explanatory scientific theory, and it's the only viable theory we have to picture the multiplicity of genetic variation and diversity shown by living things in nature. 
(1) However in "The Variation of Animals and Plants Under Domestication" (1868), Darwin - also - suggested "pangenesis" as a potential driver of evolution (over the short-term). According to the theory, beneficial traits acquired during the lifetime of an organism are passed onto offspring, and particles called "gemmules" shed by body cells could become localized in the reproductive organs. Thus - to Darwin, evolution could be pushed for a short span of time - a few generations by pangenesis.
"The modifications acquired through Sexual Selection are often so strongly pronounced that the two sexes have frequently been ranked as distinct species, or even as distinct genera. Such strongly-marked differences must be in some manner highly important; and we know that they have been acquired in some instances at the cost not only of inconvenience, but of exposure to actual danger."
"The development, however, of certain structures - of the horns, for instance, in certain stags - has been carried to a wonderful extreme; and in some cases to an extreme which, as far as the general conditions of life are concerned, must be slightly injurious to the male. From this fact we learn that the advantages which favoured males derive from conquering other males in battle or courtship, and thus leaving a numerous progeny, are in the long run greater than those derived from rather more perfect adaptation to their conditions of life." -- Darwin (1871)
"Finally, if my chief conclusion is correct, and if the neutral or nearly neutral mutation is being produced in each generation at a much higher rate than has been considered before, then we must recognize the great importance of genetic drift due to finite population number in forming the genetic structure of biological populations. The significance of random gentic drift has been deprecated during the last decade. This attitude has been influenced by opinion that almost no mutations are neutral, and also that the number of individuals forming a species is usually large that random sampling of gametes should be negligible in determining the course of evolution, except possibly through the "founder principle". To emphasize the founder principle but deny the importance of random genetic drift is, in my opinion, rather similar to assuming a great flood to explain the formation of deep valleys but rejecting a gradual but long lasting process of erosion by water as insufficient to produce such a result." -- Motoo Kimura (1967)
(4) Required reading: Professor Handford's - "A Brief History of Evolutionary Genetics": http://instruct.uwo.ca/biology/Zoo441a/hist.html