First, a quick lesson… Monstera is a genus in the family of Araceae - the family generally called “Aroids.” [To add to the confusion of classifying plants, there is a subfamily in Araceae called Aroideae. The Aroideae subfamily includes the popular genera Aglaonema, Alocasia, Dieffenbachia, and Philodendron. Monsteras are in a different subfamily - Monsteroideae. But when someone refers to "Aroids" they usually mean the whole family of Araceae (not just the Aroideae subfamily). Therefore they are referring to all subfamilies, including the genus Monstera in the Monsteroideae subfamily. The Monsteroideae subfamily includes 12 different genera, including: Epipremnum (pothos ivy), Rhaphidorpha, Scindapsus (S. pictus) and Spathiphyllum (peace lilly).]
I only gave that little lesson to continue to ingrain it in my head. I am not a botanist by trade, but I’m very interested in classification and try to keep these distinctions as I talk about plants.

Monstera deliciosa v. 'Borsigiana' - I wish this photo hadn't turned out so yellow...
Common names for the Monstera genus include “Mexican breadfruit” and “Swiss cheese plant.” The Monstera genus contains 22 species (according to wikipedia) and a number of naturally-occurring varieties and cultivars (human-cultivated varieties). I have 7 of these plants from 5 different species/varieties:
Monstera adansonii

my young Monstera adansonii
This plant was sent to me a couple of months ago by my friend, aroid collector Russ Hammer. It is one of the vining types of Monstera with a small leaf initially. The two leaves on my plant are just about 5 inches long. If my plant were to reach full maturity (unlikely), the leaves could become 3 to 4 feet in length. Here is a picture of Russ’s mature specimen (notice the size of the leaves in comparison with the hand):

mature Monstera adansonii
Monstera deliciosa ‘Borsigiana’

Monstera deliciosa v. 'Borsigiana'
Monstera deliciosa is the most common Monstera species, usually kept as a houseplant. Sometimes it is mislabeled as a split-leaf Philodendron. While the name “split-leaf” is perfectly applicable as a common name, the name Philodendron should only be applied to species that are in the Philodendron genus. [Way too many plants are called Philodendrons by common name that are not in the Philo genus.]
I have had this plant just about a year now and it has grown tremendously. I’m not sure where I’m going to put it next winter if it grows as much over the summer as it did this last year! This is a gorgeous plant. I’ll probably just have to start separating it into a couple of pots and give some away. I guess I could also experiment a little, starting a plant as a climber. So far, my plant has just been a huge bush. But this plant really likes to climb, so I guess I could give that a shot if I decide to divide it next year.
Monstera deliciosa ‘Borsigiana’ variegated

Monstera deliciosa v. 'Borsigiana' variegated
This is another plant given to me by my friend, Russ. It is a variegated variety of the plant above. The variegation is variable, from solid white leaves to solid green leaves and everything in between. One of my leaves is almost just perfectly half and half. My plant didn’t get much time outside before the weather turned cold and it had to be brought indoors. So right now it is a fairly small specimen with just 4 leaves. I expect it to really take off next summer, though. I’d like to see this turn into the “monster” that my other deliciosa has become.

Monstera deliciosa v. 'Borsigiana' variegated
Monstera obliqua (aka Monstera friedrichstahlii)

Monstera obliqua / Monstera friedrichstahlii
It seems that both of these names (obliqua and friedrichstahlii) refer to the same species of plant. The name friedrichstahlii seems to have fallen out of use recently. This species also closely resembles the species adansonii. The differences are more apparent when the inflorescence (blooming stage) and infructescence (fruiting stage) are observed. There is a discussion on the Aroid forum that may be of help here. I haven’t been lucky enough to see an inflorescence on any of my Monsteras yet, so I have to take someone else’s word on the identification.
Monstera standleyana

Monstera standleyana
This is a unique, climbing species of Monstera that is most frequently (as a houseplant) seen growing on a totem. It can become very dense, with the leaves closer on the stem than some other species. The leaves have some dappled variegation, as if some paint was dripped on them.
This is another plant that was given to me by Russ. I potted it up as soon as I received it, but it started looking bad about a week later. The leaves curled up into little scrolls and started to turn dark. I unpotted it and found that most of the roots were black and mushy. I pulled away the rotted roots and stuck the firm clipping in water in the windowsill. The cutting has been slow to put out new roots, but the leaves look better, having unravelled and flattened out. I plan to just leave the clipping in water until the Spring, at which time it should have enough roots to be put in dirt again and support itself.
New growth on my Monsteras
Unlike most of the plants I have to bring indoors for the winter, my Monsteras continue to growth. Here are a couple pictures of the beautiful new growth that I have been watching over the last couple of weeks.

new growth on small Monstera deliciosa v. 'Borsigiana'

new growth on Monstera obliqua
In nature
Monsteras are found in central and tropical south America climbing up the trunks of trees in rainforests and jungles. They are epiphytic, meaning they will attach themselves to trees. However, epiphytes do not harm the host tree, as parasitic plants do. Monsteras display heteroblastic growth, changing leaf shape as they mature and climb higher into a tree. The fenestrations (holes in the leaves) are one of the most distinct attributes of the Monstera genus, though fenestrations do occur in other genera. Actually fenestrations only occur in the Monsteroideae subfamily, a couple of other Aroid genera and an aquatic plant Aponogeton madagascariensis (Madagascar lace leaf plant). I have actually kept the madagascar lace leaf plant in my 29 gallon aquarium before. It is a very usual and interesting plant. For tree-climbing aroids, these fenestrations help the plants to weather wind in the upper branches, and prevent them from being pulled off the tree. And of course, they are the reason for the nickname “swiss cheese plant.”
Care
Monsteras don’t require any special sort of care. Generally, I keep mine in bright, indirect light. This means that they sit on my partially shaded (dappled light) back porch during the summer and are very happy. Otherwise they are in as much light as I can find inside the house during the winter. My M. obliquas usually have a couple of leaves turn yellow and fall off whenever I bring them indoors. I guess they just go into shock with the decreased light, because it has happened each year. The vining types (adansonii, obliqua) are very easy to propagate by serpentine layering or simply sticking cuttings in a glass of water in the windowsill.
As I mentioned in a previous post, I have propagated my M. deliciosa by sticking a leaf with attached stem into a glass of water, too. I was surprised to see roots grow from the base of the stem, since the piece didn’t even have a single node on it. But it sent out roots in a matter of days!
If you’re interested in seeing some more Monsteras, there are lots of good pictures of Aroids over at http://aroidpictures.fr. Here’s a direct link to the Monstera pics.