An allotetraploid refers to an organism that contains four complete copies of the genome, but two (and on rare occasions, one) of the copies are from a different species than the other two copies.

That makes no sense. Explain this to me in more detail, please.

Each living organism is described genetically by a particular number of chromosomes; these chromosomes are contained within each cell that makes up the organism. You can think of a chromosome as a piece of a blueprint, and the overall set of chromosomes which would make up a complete "blueprint" is called the genome. A living cell with a single complete genome is called a haploid cell. Most organisms contain two complete copies of the genome for redundancy purposes; these organisms are called diploids; humans are diploids.

Tetraploids, then are organisms that contain four complete copies of the genome in each of their cells.

Usually, organisms with duplicates of the genome usually have duplicates that are pretty much the same as the original. When this is the case, the prefix auto- is added to the description of the genome duplication of the organism; for example, humans could be described as autodiploids.

However, in some unusual cases, the genome duplications are significantly different from each other; different enough that they would probably be considered different species if they were not in the same cell. In this case, the prefix allo- is added to the description of the genome duplication of the organism.

So, an allotetraploid is:

allo - an organism with genomes that are quite different from one another
tetraploid - an organism with four complete genomes
or, in summary, an organism with four genomes in which some of the genomes (usually two) are significantly different from one another.

OK, now I understand what an allotetraploid is, but how can such a thing happen?

In the process of "normal" evolution, species change over time. Species might evolve slowly into something different, or might branch off into several species (like humans and chimpanzees). This branching is due to natural selection, in that a group of organisms that are best equipped to deal with an environment will eventually outnumber the members of a group that are not as well equipped.

Normally, species that are significantly different from one another cannot produce offspring (for example, humans and dolphins cannot have children), or if they do, the offspring are disadvantaged and usually sterile (mules, for example).

An allotetraploid is the result of an exception to this rule. It often occurs when a species diverges into multiple species, which occurs when a species is very widespread and begins to adapt to specific environments, eventually accumulating enough genetic difference from other portions of the original widespread species to be unable to intermate. An example of this will follow, if you're not sure of what is meant by that phrase.

What will happen is that the attempted intermating of two diverging species will result in both species retaining their full genome in the offspring. Since most species are diploid, each contributed genome will then duplicate itself in the first cell of the new organism right after fertilization, almost as though the other genome isn't even present.

In most cases, the resulting confusion will result in death, but in a few unusual cases, the organism will survive and take on the form of a mixture of both species. In very rare cases, the organism will be viable, and sometimes it will thrive.

Allotetraploids can often be made in a laboratory as well by inserting whole genomes of distinct species into freshly-fertilized cells.

So, there are really allotetraploids that exist in the wild?

There are several allotetraploids, but the two that have been studied in detail are Nicotiana tabacum (i.e. tobacco) and Zea mays (i.e., maize or corn), with maize being more widely documented.

Tell me how, for example, maize became an allotetraploid

Maize is a member of the general family of plants known as the grasses, which became widely spread about 60 million years ago. They quickly became one of the dominant plant families, spreading over the entire globe.

Eventually, the grasses began to adapt to particular climates. The rice plant became very widespread in Asia, and the wheat plant evolved in Europe, for example. The general line of grasses that developed into maize were in the mountains of South America. This general group of grasses evolved to be quite large, some as tall as four feet.

From this group came two plants that are very similar in nature to one another, maize and sorghum. These two plants began to diverge about 20 million years ago, as sorghum spread into the moist climates of Brazil, while maize stayed in the drier regions of the continent.

At this point, maize was a diploid that looked something like a miniature version of what the corn plant looks like today. After 5 million years, it had essentially become its own species, distinct from sorghum, and produced cobs covered in kernels much like today, except with much smaller and less digestable kernels. At this point, an event happened somewhere in the lower Andes in South America.

An attempted intermating between a small group of maize plants and a small group of sorghum plants resulted in the sorghum genome (at that time... the sorghum plant today is much different than its progenitor 15 million years ago) and the maize genome appearing together in the fertilized cell. Somehow, this allotetraploid managed to survive and even thrive. The resulting maize plant had much larger kernels that were sweeter and tastier and more suitable to animal ingestion; animals would then spread the seeds about, allowing maize to spread.

Maize spread throughout South America, while the version of maize that didn't intermate with sorghum, which is called teosinte, spread up into Mexico and the southwest United States. Eventually, the Incas, Mayas, and Aztecs began to domesticate the maize plant, and the result is the crop that most of us eat today.

How could allotetraploids be useful in a laboratory?

Labs might make their own allotetraploids to help determine causes of and treatments for species-specific diseases. For example, creation of an allotetraploid version of different species of mice has resulted in better cancer treatments.

Another example of how allotetraploids might be useful is the development of better food crops, in which wholesale traits from one species are added to another. An example of this is in citrus, where various citrus varieties are created in such a fashion. Another less real example is that of the tomacco plant from The Simpsons.