Genome is the total amount of DNA present in the nuclei of the cell
and DNA is the polymer of nucleotides - Adenine, Thymine, Guanine, Cytocine. Some DNA present in the cell is also present in the form of extra-nuclear form called as plasmid.
DNA (precisely nucleotides from which it is made up of) is the basic unit of code of life and Genome is the total amount of that code. Some DNA present in the form of plasmid is not included in the code.