Genes are what life forms use to pass on traits to their offspring. Genes are stored in DNA (and, to a lesser extent, in RNA), which in turn is stored in chromosomes.
Genes are determined by the sequencing of five different chemicals- adenine, thymine, guanine, cytosine, and uracil (though the latter only exists in RNA, and replaces thymine). The way these different chemicals are organized (in sets of three at a time, known as codons) determines how DNA and RNA interact with one another, which in turn determines which amino acids in our bodies are synthesized and what resulting proteins are created. These proteins then determine how our cells behave, which influences our behavior at the macroscopic level and the state of our bodies.
Gregor Mendel, an Austrian monk and scientist living in the 1800's, was among the first, and is the one credited with, discovering the existence of genes. He did so by breeding pea plants to one another and studying the resulting appearance and frequency of certain traits in their offspring. He compiled quite a bit of data to support his findings and hypothesis, and though his religious peers and the general public viewed his work as somewhat heretical and insane at the time, today, Mendelian Genetics is accepted as a fundamental fact, even by many religious officials. (Ironically, Charles Darwin put forward his Theory of Evolution by Natural Selection at nearly the same time, to similar and ultimately greater backlash, and admitted that one of the then-flaws with his theory was that a system of inheritance which would allow for natural selection- exactly what Mendelian Genetics delivered- had yet to be discovered.)
Effects of genes
The existence of genes allows both for the passage of traits and their corruption via mutations. Mutations then allow for genetic diversity and can (and do) lead to evolution, both on a micro- and macroscopic scale. The result is the existent and still continuing emergence of biodiversity, as well as common descent between species.
Genetics is the study of the genome.