An allele is a different variation of a gene.  Genes are the subunits on chromosomes that cause the expression of a particular trait, or the absence of a particular trait.  An Austrian monk named Gregor Mendel pioneered the work in genetics by working with his pea plants in his garden.  Mendel kept careful notes on his pea plant experiments.  He noticed, for example, that purebred tall pea plants always produced tall offspring, and purebred short pea plants always produced short offspring.  One day he decided to cross a tall with a short; the result was all the offspring were tall.  Then, he took two of those offspring and crossed them.  Much to his surprise, three were tall, while one was short!  Today, we know this is the result of mixing different alleles for plant height.  In this experiment, the tall plants were dominant, we would label the allele for tallness with a capital "T", while the short plants were recessive, so we would label the allele for shortness with a lower-case "t".  The first cross would look like this, for a tall versus a short:

            t          t

T         Tt         Tt

T         Tt         Tt

100% of the offspring had the allele for tallness, so they were all tall.  The second cross would look like this:

            T         t

T          TT       Tt

t           Tt        tt

In this cross, we see the reemergence of the two recessive alleles, which would produce a short plant.

What is allele?

An allele is a variation of a gene.  Genes come in pairs, and they are located on homologous chromosomes.  

Let's work with eye color for examples.  I realize that eyes come in a variety of colors, but for the sake of simplicity we'll say that eye color is either brown or blue.  A person receives a set of genes for eye color.  One eye color gene (located on a chromosome) came from mom, and another eye color gene (located on the matching homologous chromosome) came from dad.  Alleles are the specific version of those genes.  

The gene is eye color, and it comes in two variations -- brown or blue.  One allele could be for brown eyes.  The other allele could be for blue eyes. Either way, we are still talking about eye color.  

Both dad and mom could have donated brown eyed alleles.  Both mom and dad could have donated blue eyed alleles.  Or one parent could have donated brown, while the other parent donated blue.  Once you have different alleles, the trait that is shown comes down to a question of which allele is dominant.  If I were to inherit a brown eyed allele and a blue eyed allele, my eye color would be brown, because the brown allele is genetically dominant over blue.