I work mostly with laeliinae, so I'm going to give examples from such.
In normal reproduction, the gametes (sex cells) are haploid (1n) and this in the case of a normal Cattleya means the sperm and egg cells have 20 chromosomes. THis is an artifact of meiotic division, which normally splits all chromosomes in half. This is the basis of genetic variation, because specific genes that affect the phenotypic expression (what you see) of any organism exist in specific places on the chromosomes. If the part of the chromosome that ends up in a sperm or egg cell does not have a particular gene for colour or something, that egg or sperm cannot contribute that gene, but the other cell that divided off from the one in question would.
Now, in some cases, meiosis does not go as expected and you end up with sex cells that have a full compliment of genes from the parent. Now, instead of haploid (1n; 20 chromosomes in Cattleyas) gametes, you have diploid (2n) gametes. In most animals, this sort of mutation is deleterious, but in plants, polyploidy is often not at all a problem. So, if you have an ovary full of diploid egg cells, or even just one, and a regular haploid sperm cell comes along and merges with that egg, you end up with offspring that have one full set of chromosomes from one parent and half of the chromosomes of the other, which is called triploid (3n). Triploids, as it turns out, are sterile, usually. Meiosis cannot proceed properly with an uneven division of chromosomes. If, by chance, a triploid should produce a viable egg or sperm, it is likely to be a normal haploid cell, or maybe a diploid cell, which will breed as above. Triplods often have many of the cosmetic advantages of tetraploids (larger, better-substanced flowers, stronger growth), but are terrible breeders. Some famous examples are C. warsewiczii 'F. M. B.', perhaps one of the best alba warsewicziis ever, and C. trianiae 'Aranke Germaske', another alba. Both lovely, both absolute
sh*t for breeding.
Now, should an egg cell have a diploid chromosome count (in the case of most Cattleya alliance plants: 40) and a sperm cell comes along that also has a diploid chromosome count, you get an offpsring that is a tetraploid (4n). In Cattleyas, a tetraploid has 80 chromosomes. Now, normally, this means that you will be looking at larger, heavy-substanced flowers with better than average shape and that tend to last longer on the plant. Tetraploids also tend to be stronger, if slower, growers (takes more time for those chromosome-heavy cells to divide) and tolerant of a wider range of conditions than are their diploid peers. These are all positive things, from a horticultural point of view. Under normal circumstances, when meiosis occurs in a tetraploid, the resulting gametes have 40 chromosomes, just as a normal diploid meiosis results in 20 chromosomes to a gamete.
Hope this helps! Feel free to ask for further clarification!
-Cj