DNA: It is the code where it is written the genetic information. Only four different letters A, T, C, G (the nitrogenated bases adenine, thymine, cytosine and guanine) alternate many times in the DNA attached to a double helix backbone of sugar and phosphate.
CHROMOSOME: The DNA it is a very long molecule that coils and packs in the form of chromosomes. The human being has 23 pairs of chromosomes (each chromosome is duplicated, coming one from the mother and one from the father).
GENOME: It is all the information needed to build and maintain an organism. This information is hereditary and encoded in the DNA.
GENE: The genome information is divided into discrete units called genes. Each gene has the information to build a molecule that has a function in the organism. The human being has about 30.000 different genes on its genome.
And now, after these quick definitions, lets make it simple with an example:
Lets imagine that you have a huge encyclopedia on “how to build a human being”. This encyclopedia it is your genome and each article it is a gene. Then the chromosomes would be represented by the different books that make up the whole encyclopedia, being the DNA the paper (sugar and phosfate backbone) and the printed letters (nitrogenated bases that encode the information).