In biology, one of the two most common and best understood forms of altruism. The other is reciprocal altruism. Kin altruism occurs because close relatives (kin) have a higher probability of sharing genes than random members of the population.
Obviously, parents take care of or invest in their children. Even parents who are no longer there when their eggs hatch, have provided their offspring with nutrition in the form of the egg. This is so obviously necessary for the perpetuation of the lineage that this parental altruism is often not regarded as altruism; and the term kin altruism is sometimes restricted to other forms of investment: in younger siblings, nieces and nephews, grandchildren, and so on.
But, although there are particular reasons why the parent-child relationship is slightly different from other forms of kinship, the genetic basis is essentially the same in all cases, and the same effect is occurring.
In normal sexually reproducing organisms, parents and children share half their genes; as do full sisters and brothers. More distant relatives share proportionately less. Your genes are likely to benefit if you help your sister; less so if you help your cousin. "Your" genes are not just those copies in your body, inherited from one or other of your parents, but the same gene in anyone else who has inherited it from the same source.
In all this, though I'm using human kin terms, it does not really apply to humans, because we have too many cultural reasons to behave altruistically or otherwise, and culture develops faster than biological evolution. However, the effects of kin selection are still sometimes detectable: it is reported that stepchildren are killed more often than blood children.
Natural selection that occurs as a result of the good done by kin altruism is called kin selection. This is occasionally erroneously (or at least very unhelpfully) classed as a form of group selection, that is where changes occur for the benefit of a group (a family, a species, whatever). It is generally agreed that group selection in this proper sense does not and cannot occur.
Most sexual organisms are diploid, having chromosomes in pairs, one member of each pair coming from each parent. This gives the coefficient of relationship of 1/2 between parent and child, or between full siblings, 1/4 between half siblings, and so on: the probability that any given gene in one body will also be in the other body, because of their common descent.
A haplodiploid creature has different sexual arrangements, and this typically leads them to being eusocial, like bees and ants, with a single fertile female and a sterile caste. The sterile creatures don't want children of their own because they are more closely related to their siblings than they would be to their own children. So it's to the advantage of their genes to work for or guard the hive or nest, and not breed. It is still kin altruism at work, but with different probabilities. (See Sister power! Close relationships and hymenopterans for a more detailed discussion.)
Organisms do not practise kin altruism because it is good for the other individual they are benefiting. The other individual benefits, but that is not the reason for it. Nor does the organism do it for selfish reasons, i.e. to benefit itself. By definition, altruism is a direct loss of benefit to itself - though it may gain by later return of the favour: but if it does, this is the unrelated phenomenon of reciprocal altruism, which has a different genetic basis.
There is no intrinsic reason to favour your kin as such. The reason they're valuable to your genes is that they are likely to contain your genes. One gene prospers when it helps many copies of itself arise and survive. One way of doing this is by causing your body to have many offspring, each of which is likely (probability = 0.5) to contain you (the gene).
Another way is to be able to recognize other bodies that are likely to contain you. And one good bet on that count is if you can recognize kin in general. One good way of doing that is if you behave altruistically to individuals brought up with you. These are all probabilistic devices for taking a good guess at who shares your genes. Cuckoos exploit this by the fact that usually the thing in a mummy bird's nest is mummy's baby bird, so mummy is usually on the right track genetically by feeding any bird in its nest.
Another way of recognizing kin is by clusters of features: how much they resemble you in looks, smells, voice, or whatever. If you're a gene in a body that closely resembles another body, you have no particular interest in fostering the other genes that cause those, but they mark the likelihood that the body is of recent common descent with you and therefore could contain you.
Having (say) red hair, and being altruistic, are two entirely different effects of genes. It is possible that one gene could have two separate effects, (i) giving you red hair, and (ii) making you behave altruistically to people who giggle a lot. One gene having two unrelated effects is called pleiotropy. But it is vanishingly unlikely that a single gene could have the two pleiotropic effects of having red hair, and behaving altruistically to people with red hair. The chemical pathways for these effects are totally different. Richard Dawkins called this hypothetical coincidence the green-beard effect. Years later an example was reported in nature. [Don't click on the link: I haven't noded it yet, and it's pretty much explained here.]
But normally it's impossible to recognize that another individual actually bears the same gene for altruism as you do: so if you behave altruistically to those most likely to bear it (your kin), on average the gene for altruism will be helped to survive. So all the other genes in the helped individual's body will also benefit.
Further reading: The Extended Phenotype (1981) by Richard Dawkins contains vastly more detailed discussion. His much more accessible The Selfish Gene (1976, revised 1989) helps me understand the other book.