A gene is part of the DNA strings that can be found in ech cell's nucleus, the gene is a code for how to build a protein, that is it encodes the order of the amino acids. The proteins in turn can have one of several function in an organims, and so determines partly what the organism is.
The actual procces that makes the proteins is consists of two steps. The first step is called transcription, in this step the DNA is transcribed to RNA. The second step is called translation, in this step the RNA is translated to a protein. Transcription takes place within the cell's nucleus, translation takes place in the cytoplasma with the help of ribosomes.
A protein is a chain of amino acids joined by so-called pepetide bonds. Parts of a protein or a whole protein is often called a polypeptide. The order of the nucleotides in the DNA chain determines the order of nucleuotides in the RNA and this determines the order of aminoacids in the protein. So the different order of the nucleotides in the different genes makes them code for different proteins.