Alleles are different versions of the same gene. In popular every day usage, when people talk of "a gene for something" they often mean what is correctly called "allele". There is no gene for blue eyes, for example, or gene for wrinkled pea seeds: what exists is a gene for eye color (in fact, more than one) or a gene for the shape of the pea seed.
In classical genetics, a gene was understood to be a single unit of inheritance, responsible for appearance of a particular trait in the living organism. An allele is variant of a gene, determining what form the trait will take in the actual living organism (what is in genetics called phenotype).
In modern molecular genetics, a gene is understood as a length of the DNA sequence that carries a particular instruction, or a set of instructions (often but not always related to production of particular protein), located in a specific position on a particular chromosome. This position is called a "genetic locus".
Alleles are now understood to be variant DNA sequences in the same genetic locus. Alleles are elements of the genotype. They might, or might not, result in appearance of variant phenotype characteristics.
Each diploid living organism (including most mammals, many other animals and some forms of most plants) has two copies of most genes: one inherited from the mother (originally present in the egg) and one inherited from the father (originally present in the sperm cell).
An egg fertilized by the sperm forms an initial cell from which embryonic development starts, called a zygote. All the other cells of resulting embryo, fetus and the final living organism are form by divisions of this initial cell, and carry the same set of genetic information (save any mutations that might occur during the development). Thus, as a kind of shorthand, a whole organism can be described as a homozygote or a heterozygote from the point of view of any single gene. A homozygote carries two identical alleles of the gene, while a heterozygote carries two different alleles of the same gene.
A shape of a pea seed is determined by a single gene, and has two phenotype versions: the seed can be round or wrinkled. The gene for the seed shape has two alleles, conventionally written down as R (round) and r (wrinkled). Each plant thus can inherit one of the three allele combinations: RR, rr, Rr. RR and rr combinations are homozygotes, and the rR is a heterozygote.
Many genes have more than two alleles, some significantly more. The gene for the main human blood group exists in three alleles: A, B and O resulting in six possible genotypes: AA, AO, BB, BO, AB and OO. Due to the genetic dominance patterns of the blood group alleles, these six genotypes result in four phenotypes that can be found in the human population (blood groups A, B, AB and O).
The term dominance describes a situation in which one allele (encoding the dominant trait) has a stronger effect than another (encoding a recessive trait) in the phenotype. In the case of pea plants, the round seeds are dominant over the wrinkled seeds and thus a heterozygote with a round allele and a wrinkled allele (Rr) will appear identical to the homozygote with two round alleles (RR), and a genotype with two wrinkled alleles (rr) is necessary for a plant with wrinkled seeds to appear. Similarly, A and B blood groups are dominant over the O group, and thus heterozygotes AO and BO are in the phenotype indistinguishable from homozygotes AA and BB.