From smallest to largest: nucleotide, gene, chromosome, genome.
Nucleotides are the smallest building blocks of DNA. There are four nucleotides (A, G, T, C) which arrange in pairs to form the long double strands typical of DNA molecules.
A gene is a segment of DNA which codes for the amino acid sequence of a particular protein. A gene is therefore composed of many pairs of nucleotides.
A chromosome is a long strand of DNA which is coiled up with various proteins. A chromosome contains many genes.
The genome is all the DNA of a particular organism. All of an organism's chromosomes compose the organism's genome.
To summarize, we can think of the genome as being composed of a collection of chromosomes, of chromosomes as being a collection of genes, and of genes as being composed of nucleotides.