When an organism reproduces, it passes on some of its attributes to its offspring. In asexual reproduction, offspring are identical copies of a single parent. Each child is a clone -- its genes are identical copies of its parent's genes. If the child of an asexually reproducing organism is different from its parent, it is because a mutation has occured.
In sexual reproduction, offspring inherit a mixture of attributes from two parents. In this case, in individual's genome is actually two strings. One string is a set of genes inherited from the mother, and the other string is the set of genes inherited from the father. An allele is thus identified by a pair of letters; for example, if one parent contributes a gene of type A and the other a gene of type a, the allele of the child is Aa at that locus.
Sexual reproduction is based on a process known as recombination. From each parent's double stranded genome a single strand is created by randomly selecting one of the two genes at each locus. The child's gene will consist of one single strand from the mother and one single strand from the father. For example, a parent of type Aa at a particular locus could contribute either an A or an a to its offspring at that locus. If one parent is of type Aa and the other of type Bb, there are four different possibilities for the child: AB, Ab, aB, and ab.
Sexual reproduction gives a species an advantage over asexually reproducing species: in sexual reproduction, any given individual can be healthier than either of its parents. For example, suppose that in the original population all individuals had the allele AA at a certain locus, but after some number of generations many were of type Aa, where a is a slightly harmful mutated gene. Now consider the children of two Aa parents. Some may well be aa, and thus less fit than their parents. But others could be AA, inheriting the best genes from both parents, and these children are healthier than either parent.