Most discussions of genetics demonstrate that what is known has been discovered through rigorous scientific experimentation and observation. One good reason is that this field demonstrates more clearly than the study of fossils the evolutionary basis - and common roots - of all life on earth. I don't plan to go to that level of detail: because anybody who investigates this subject with any sort of rigour would find the evidence clear, logical, and irrefutable.
Most discussions also begin with Mendel's two laws. Although they are widely known, I'll reproduce them briefly here. Mendel was a monk (and scientist) whose work on breeding peas was largely ignored in the 19th century, and rediscovered at the start of the 20th.
Mendel's first law: an individual inherits two factors of heredity for each given trait, one from each parent.
Mendel's second law: independent traits assort independently of each other.
From Mendel came the understanding those two factors above can be dominant or recessive - ie the dominant can mask the recessive, although both factors are present in the individual, and inheritable.
Now, a molecular biology overview.
Animals and plants are eukaryotes: each living cell in every organism has a nucleus, which contains the genetic blueprint for the organism. This is organised as a set of chromosomes, the same number within each species, but differing numbers for different species.
In contrast, Bacteria are prokaryotes, which means a bacterial cell has no compartmentalised nucleus for the storage of genetic material. Bacteria are single-celled organisms, as are nearly all prokaryotes. Most eukaryotes are multi-celled organisms, although some are unicellular - amoeba being an example.
(Although it could be said that organisms lacking nuclei are more primitive, Stephen Jay Gould has illustrated that the likes of bacteria are an extraordinarily successful form of life, having eked out more niches on this planet, and over a longer period of time, than anything else. Moreover, bacteria in total biomass outweigh the total of all eukarytic life. They are very successful adaptors.)
Chromosomes take the form of DNA (deoxyribose nucleic acid): very lengthy molecules - largely comprising hydrogen, carbon, oxygen and nitrogen - that are typically packed tightly, in a well-structured way. Much of a cell's metabolism is controlled automatically via the instructions stored in DNA, which are sent out from the nucleus through RNA, and typically executed by the assembly of proteins from the building blocks of amino acids.
Proteins typically act as catalysts: that is, they facilitate chemical reactions without being changed themselves. Thus, the presence of particular proteins bring about certain reactions that affect metabolism in cells and through the body.