And how are they related to DNA, exactly?
A gene is a section or a stretch of your DNA sequence. Its role is to produce proteins and other molecules that humans need in order to function. Genes can be hundreds or millions of letters long. And by letters I mean A, C, G or T, which are known as nucleobases.
A genome is the complete set of DNA, including all the genes. Genes are just 1-5% of your genome. The rest is used to regulate the genes.
Hope this clears it up.