An allele is a particular manifestation of a gene, which was called a factor by Mendel. Different alleles determine variations of an organism's traits. Some of these traits can be observed (height, pigmentation, etc.), whereas others are hidden (metabolism or predisposition to disease).
Most multicellular organisms, including humans, are diploid. Thus, every gene is present in two alleles, one for each chromosome in the homologous pair. Alleles are physically encoded on the chromosome by way of small variations arising from mutations; if the two alleles for a given gene are identical, then the organism is called homozygous for the gene; otherwise, the organism is heterozygous.
If the heterozygote is indistinguishable from one of the homozygotes, the allele common to both organisms is said to be dominant over the second allele, which is called recessive. Thus, the dominant allele masks the expression of the recessive allele.
Actually, most observable characteristics of multicellular organisms are non-Mendelian traits, meaning that they arise as a result of the interaction of multiple genes, and so the simple picture of two conflicting alleles gets muddled.
Examples of human traits that are Mendelian are the ability to detect the almond smell of hydrogen cyanide (recessive), albinism (recessive) and even wet or dry earwax (dominant and recessive, respectively).