A monster's Type is a family or grouping almost like a genus name. This type of classification came into play in Monster Rancher Advance 2 to help breeders with combining powerful breeds of monsters. In that game, they made it clear that Sprites could not be combined with Fancifuls, Artificials could not be combined with Beasts, and Spirits could not be combined with other Spirits. Obviously other games ignored this rule.
The 5 types are as follows:
These monsters are the most like wild animals or "pets." They are generally easy to handle and can be great starting companions.
These are "created" monsters that can sometimes be difficult to understand. Many of them are made of nature elements or built by human hands.
Strange monsters whose origins are often shrouded in mystery. Many of them seem like otherworldly creatures and they can make odd-looking monsters when combined.
Often some of the easiest monsters to handle, and sometimes deemed the "cutest" monster group because many of them are rather small.
Some of the most powerful monsters reside in this group. Many are even more intelligent than humans, so training them can be a chore and at times even dangerous.
Obviously, since these don't all appear in every game, some of this is just speculation. However, every breed stays the same even after combined. For example, all Maya breeds are Sprites and all Dragon breeds are Spirits.