First, let us clearly establish what a "Gene" really is. Each of us is made up of millions of individual cells. Cells make up your skin, cells make up your intestines, heart and other organs, and cells make up your muscles, your bones, and your brain. All of the cells in a given part of the body are more or less similar, while cells that make up different parts of the body are more or less different. One thing all cells in all animals (in fact all forms of multi-cellular life and a good proportion of uni-cellular life) have in common, is a nucleus containing long complex molecules of "deoxyribonucleic acid", otherwise known as DNA. In repeatable experiments, biochemists have proven that the exact chemical structure of these DNA molecules is what controls the shape, nature, and function of the cell that contains them. Small segments (I shall call them "Steps", although actually each "Step" is really a set of three base-pairs linked to the sugar backbone) of the DNA molecules can be structured in a quite limited number of well-understood ways. But these "Steps" can be chained together into very long strings. These long strings of DNA are also called "chromosomes".
Now consider a given set of "Steps" in a particular chromosome. And lets assume, for a moment, that the particular chemical configuration of this particular set of "Steps" controls some chemical process within the cell. This particular set of "Steps" is called a "Gene". Lets call this very specific set of "Steps" on this very particular chromosome "Gene A", for the purposes of further discussion. Any collection of "Steps" in a chromosome that controls or influences some chemical process in the cell, is called a "Gene". Biochemists do not yet fully understand the workings of Genes, but they have properly proven a number of their characteristics that are critical to this discussion. Most importantly, they have proven that certain particular groupings of steps in the DNA molecule exert critical control over the functioning of the individual cell, and over the organism as a whole. A number of human and animal diseases, as a prime example, have been shown to be the direct result of an "error" in the sequence of steps that constitutes particular "Genes". The scientists have also shown that not every "Step" in a chromosome is part of a Gene. It would appear that there are long stretches of "Steps" that serve no discernible purpose. These "introns" (as the specialists refer to them) appear both outside of known genes, and within the step sequence of genes. And, of course, not every random sequence of "Steps" is a Gene. The processes of Evolution have spent over a billion years selecting just those particular sequences of "Steps" that are meaningful for a given species.
All this information becomes really important, when you remember that in the processes of human procreation, one cell from the male (called the sperm) and one cell from the female (called an ovum) get together to form one cell of the baby. Each of us starts as a single cell containing 23 chromosomes that we have inherited from our father, and 23 chromosomes that we have inherited from our mother. So it is these 46 chromosomes, and the Genes that are contained in these chromosomes, that is almost all that is passed (chemically) from one generation to the other. (We will ignore for the purposes of this discussion the DNA that is passed from the mother in the other organelles of the ovum since it forms a relatively minuscule portion of the total DNA involved.) It is the particular chemical structure of the DNA molecules of these 46 chromosomes that determines the bio-chemical structure of the resulting individual. And Genes, by their bio-chemical influence on the individual cells of our body, exert influence over our behaviour. (This description is quite simplified, of course, so those of you who have a more in-depth understanding of Genetics, please do not get upset. This simplified description is "True enough" for my purposes here.)
Lets not get into a discussion here about how much of the Individual is determined by their chemical inheritance, and how much by experience. The "Nature versus Nurture" argument is a subject that will require a lot more detailed information that we have room to explore here. For now, lets just agree that "Some" of what constitutes the Individual is determined by the bio-chemistry of that first cell. After all, your personality is likely to be different if you develop a penis, than if you develop a vagina. You are likely to be different if you are six foot tall, than if you are only five foot two. There are many features of an individual that are indisputably determined by genetic inheritance, regardless of how the nature versus nurture discussion progresses.
Changes to the chemical sequence of a chromosome are called "mutations". There are many possible causes for mutations, including radiation, chemical toxins, and the normal biological processes that create the sperm and ovum cells. A mutation to the chemical structure of any single "Step" of a gene, creates a new form of the gene. Because some genes contain long stretches of apparently inactive "Steps", and some contain many repetitions of their active regions, not all mutations have a detectable impact. But a change to a key "Step" of a gene, will have a detectable chemical impact on the processes of the cell. Cancer, for example, is understood to be caused by mutations in certain steps of some Genes. (What causes the mutations, and which steps and which genes are involved, is the subject of massive Cancer research programs.) So let us now consider our "Gene A" as a new form of a gene that has just appeared in your body as the result of a mutation. You have just been zapped by a cosmic ray, and what used to be Gene Q, has been changed, and is now a new Gene A.
Think a moment, on the history and life-cycle of this Gene A. If this gene should appear in a cell on the back of your hand, its life time will be very short. When you die, so does the gene. The particular physical and chemical structure of Gene A will vanish with you. On an evolutionary time scale, this new Gene A will have had no impact. It will be as if it had appeared and vanished in an instant, with no legacy of its existence. But consider the life-cycle of this Gene A, if it should appear in the cells that participate in creating Sperm or Ova. Suppose that in the process of creating a sperm or ovum, Gene A gets included in the 23 chromosomes to be packaged in each sex cell.
Now, if one of these sex cells containing Gene A should chance to meet up with its counter-part and create a baby, Gene A will exist in two separate individuals. If it happens that you die (as everyone eventually does), Gene A will still exist in the next generation. The chemical mutation to a stretch of DNA that we have been calling "Gene A", has outlived its original carrier. Bio-geneticists are interested in much longer time-spans and much larger populations, of course, than the single individual and the single generation we have been talking about here. But the basic concepts are what is important, and they apply equally well to large sample populations, and multiple generations. By having multiple off-spring, you create multiple copies of "Gene A". If your off-spring have multiple off-spring, then there are even more copies of this particular "Gene A". And so on, until the numbers of generations involved becomes large, and the population of individuals becomes large.
Now lets add a few more labels to the discussion. For purposes of discussion, I will label as "P" a Gene that somehow (the chemistry is irrelevant) reinforces behaviour in its carrying packages (you, and your off-spring) that makes you (and your offspring), in its current environment, marginally more likely to proliferate copies of itself. And lets label as "N" a Gene that reinforces behaviour in its carrying packages (you, and your off-spring) that, in its current environment, marginally interferes with your proliferation of copies of itself. And, for the sake of completeness, a "Q" gene will be one that does not appear to have any impact on your propensity, or lack there-of, to proliferate copies of itself.
Many aspects of animal (and of course Human) behaviour can now be understood as consequences of the mathematics of genetic proliferation, where they were passed off as unexplained "instincts" before. The instincts against incest, existing to some degree in all species of mammals, can now be understood as genetic-level inheritance arithmetic. Over the many generations of various forms of life, the genes that coded for various ways to hinder incestuous sexual reproduction were relatively more likely to proliferate. The hard teacher of evolutionary experience labelled these genes as "P" genes. Because of some of the more detailed characteristics of genetic inheritance, incestuous pro-creation is not a good method of proliferating copies of genes. The result can be seen now in many sexually reproducing species, as a behavioural instinct that tends to drive individuals to non-family members for reproductive purposes. The three classical instincts for Self-, Off-spring-, and Species- Preservation have also become understandable as genetic-level calculations of the odds for genetic proliferation. Within the theory of Genetic Evolution, there are no longer conflicts between the three instincts. Understood as one single Genetic-Survival theory, the mother cat dying for her kittens can be rationally explained by simple mathematics. The "altruistic" behaviour of bees in a hive, or termites in a hill, are likewise reduced to the same simple non-altruistic selfish-gene mathematics.
I am not sufficiently versed in the details of the mathematics of the Theory to provide any better description. The non-technical over-view I have presented here is the best I can do for a work of this nature. If you wish further factual information, or scientific discussion of this theory, there are numerous popular and specialist works on the subject at any reference library. This overview, however, should suffice for the purposes of our further discussion, as long as you accept the basic concepts I have presented. Otherwise, a visit to the library is recommended. Some texts that are an excellent presentation of this line of reasoning are
Darwin, Charles (Edited by J.W. Burrow). The Origin of Species. Penguin Books, Ltd.; London, England; 1968. Originally Published in 1859.
Darwin, Charles. The Descent of Man (Second Edition). A.L. Burt Company; New York; 1871.
Dawkins, Richard. The Selfish Gene. Oxford University Press; New York; 1976.
Dawkins, Richard. The Blind Watchmaker. W.W. Norton and Company; New York; 1987.
Dawkins, Richard. The Extended Phenotype. Oxford University Press, Oxford, 1982; ISBN 0-19-286088-7.
Morris, Desmond. Manwatching; Harry N. Abrams, Inc., New York, 1977; ISBN 0-8109-2184-7.
Most of these works also include extensive bibliographies to guide further reading.