Many people (even some biologists) seem to be unclear about what the theory/law of natural selection actually is. Below is a diagram which explains it. The theory was independently proposed by Darwin and Wallace in their joint paper of 1858, which was published by the Linnean Society of London in August 1858 (see HERE), 15 months before Darwin's Origin of Species. It explains how adaptive evolutionary change can occur within a lineage over time i.e. a process that scientists call anagenesis. Anagenesis is the gradual evolution of a species that continues to exist as an interbreeding population. This contrasts with cladogenesis, which takes place when there is branching or splitting, leading to two or more lineages and perhaps resulting in separate species being formed. Additional theories are required to explain cladogenesis (sympatric, parapatric or allopatric speciation). Natural selection may act at various levels (genes, individuals, or (controversially) groups of individuals), but this is an additional component - not one which is intrinsically part of the theory of natural selection. In their 1858 paper, both Darwin and Wallace proposed that selection acted on individuals (see HERE), although Darwin later suggested it could also act at the group level.
Note that in addition to the "struggle for resources" (inter- or intraspecific competition) listed above, a change in environmental conditions may mean that some individuals are better adapted to the altered conditions than are others in the population. "Inference 1" should therefore read "Not all offspring that are produced survive and reproduce, because of a struggle for resources or inability to cope with changing environmental conditions (abiotic factors)".