What are dominant vs recessive alleles?

An allele is a version of a gene, for instance a gene that determines hair colour may have an allele for blonde hair and an allele for dark hair. We all have two copies of each gene because we inherit two chromosomes, one from our mother and one from our father. If someone has two of the same allele, eg blonde/blonde, we say they are homozygous for that allele, but if their two copies are different, we say they are heterozygous.
Some alleles need two copies to be "expressed" - in this example, we might need two blonde alleles to have blonde hair. This is called a recessive allele. However other alleles, in our example dark hair, only require one copy for that trait to be expressed. So if you had dark/blonde alleles, you would have dark hair exactly as if you had two dark alleles. These alleles are called dominant.