An allele is any one of a number of alternative forms of the same gene occupying a given locus (position) on a chromosome. An example is the gene for blossom color in many species of flower - a single gene controls the color of the petals, but there may be several different versions of the gene. One version might result in red petals, while another might result in white petals.
Some organisms are diploid - that is, they have paired homologous chromosomes in their somatic cells, and thus contain two copies of each gene. An organism in which both copies of the gene are identical - that is, have the same allele - is said to be homozygous for that gene. An organism which has two different alleles of the gene is said to be heterozygous. Often one allele is "dominant" and the other is "recessive" - the "dominant" allele will determine what trait is expressed. For example, in the case of blossom color, if the "red" allele is dominant to the "white" allele, in a heterozygous flower (with one red and one white allele), the petals will be red. The recessive allele will only be expressed in a recessive homozygote.
However, there are exceptions to the way heterozygotes express themselves in the phenotype. One exception is incomplete dominance (sometimes called blending inheritance) when alleles blend their traits in the phenotype. An example of this would be seen if, when crossing flowers with codominant "blue" and "purple" alleles for petal color, the resulting offspring would have violet petals. Another exception is co-dominance, where both alleles are active and both traits are expressed at the same time; for example, both red and white petals in the same bloom or red and white flowers on the same plant. Codominance is also apparent in human blood types. A gene containing the codominant pure blood type alleles "AA" and "BB" would result in a blood type of "AB".