Inclusive fitness may be imagined as the personal fitness which an individual actually expresses in its production of adult offspring as it becomes after it has been first stripped and then augmented in a certain way. It is stripped of all components which can be considered as due to the individual's social environment, leaving the fitness which he would express if not exposed to any of the harms or benefits of that environment. This quantity is then augmented by certain fractions of the quantities of harm and benefit which the individual himself causes to the fitnesses of his neighbors. The fractions in question are simply the coefficients of relationship appropriate to the neighbors whom he affects; unit for clonal individuals, one-half for sibs, one-quarter for half-sibs, one-eighth for cousins [...] and finally zero for all neighbors whose relationship can be considered negligibly small.
Inclusive fitness can be seen as an attempt to save the concept of organism-level fitness in the aftermath of the development of the idea of kin selection. The concept and its usefulness can be rather challenging to understand.
The term "inclusive fitness theory" has become a rival to the terminology used by kin selection theorists.
The term "inclusive fitness theory" was endorsed by Hamilton - the originator of the idea of kin selection. Hamilton (1975) wrote:
The usefulness of the 'inclusive fitness' approach to social behaviour [...] is that it is more general than the 'group selection', 'kin selection', or 'reciprocal altruism' approaches and so provides an overview even where regression coefficients and fitness effects are not easy to estimate or specify....and...
the foregoing discussion shows that kinship should be considered just one way of getting positive regression of genotype in the recipient, and that it is this positive regression that is vitally necessary for altruism. Thus the inclusive-fitness concept is more general than ‘kin selection’.
However, here we will use the term 'kin selection' in perference to 'inclusive fitness theory' - and treat them as synonyms.
Much is made by advocates of using "inclusive fitness" terminology of the virtues of identity - as opposed to "identity by descent". It is true that it is identity that matters in kin selection. However, in biology, most identity is identity by descent. The alternatives to identity by descent are identity by chance and identity by convergent evolution. However for complex biological objects, identity by chance would be a miracle, and identity by convergent evolution depends on convergent evolution being a very powerful and specific force - which it mostly isn't.
In the realm of organic evolution, genetic identity is identity by descent. If two stretches of DNA of a non-trivial size are identical, it is because they are descended from a common ancestor.
In cultural evolution, identity by convergent evolution is more likely - but still so rare as to be a big non-issue. Convergent evolution usually produces similarity - not identity.
The term "kin selection" is short and neat. It fits in neatly with the term "group selection". "Inclusive fitness theory" is a mouthful. It refers to an esoteric technical concept. It makes no mention of kinship or family.
In a nutshell, the terminology of "inclusive fitness theory" totally sucks.