The genome of an organism is the entire DNA of that organism. It includes all of the genetic material in the cells, spread across all of its chromosomes. Different species have different genomes, and the genome of an individual is very similar to others of the same species but not identical.
A gene is a small section of a DNA molecule. Each gene codes for a specific protein. The order of the bases along the gene is the code that decides which protein is made.
Proteins do many jobs in the body, such as making up enzymes and structures in cells. So the genes an organism inherits, together with its environment, help to decide how it develops. A chromosome is a long DNA molecule that carries many genes, and the genome is all of them put together.