DNA is a long molecule found in the nucleus (or more specifically the nucleolus) of a human cell. When it is condensed and shortened it appears, under a microscope, as 23 pairs of chromosomes.
A gene is a long sequence of DNA. This is the same thing as a portion of a chromosome. It encodes a specific protein. This means that the specific DNA sequence can be read by an enzyme and through various cellular processes (transcription and translation) a protein is synthesised. This protein may have its own function or contribute to a larger protein structure that has a function. The protein may function at the cellular level or may be involved in a visible characteristic such as hair colour.