Different alleles of a gene, called variants, all arise from mutations. A mutation is a random change in the sequence of bases in DNA. The change can happen by chance when DNA is copied, and it happens more often after exposure to ionising radiation or some chemicals. A mutation in a gamete can be passed on to the next generation.
Most mutations have no effect on the phenotype. This is often because they happen in non-coding DNA, which does not code for a protein, or because the protein that is made still works in the same way.
Some mutations influence the phenotype, for example by changing a characteristic such as height a little. A very few determine the phenotype. The allele that causes cystic fibrosis is an example, because a person with two copies of it has the disorder.