The Element a monster associates with is mostly used for classification, but becomes important in Monster Rancher 4, Monster Farm Online and Monster Farm Lagoon. Nevertheless, monsters can generally be classified by elements depending on the type of attacks they have or the traits that they give when combining.
In Monster Rancher 4, monsters aren't assigned an element per se, but their techniques are. Many techniques that do not have an element already assigned have the option of adding an element to them once your monster has picked up a trait for that element. For example, Tiger's "Bolt" is obviously already a Thunder attack, but its "Bite" does not have an element associated with it. If your Tiger picks up the trait Fire Lore, it will have a fire element added into the animation of the attack, and it will also have a special effect. This will also make the attack extra powerful against the opposing element, i.e. Fire vs. Ice. To add an element to a technique you need to go into the Customizing screen. Press the square button and move until you are next to one of the boxes on an assigned move. Press the X button and choose one of the elements you know the lore for. The elemental icon will then appear next to that attack and the corresponding effects will be added. Click HERE to see which attacks can be modified. Only 8 elements are included in Monster Rancher 4: Fire, Earth, Water, Wind, Ice, Thunder, Magic and Heart.
In Monster Farm Online / Lagoon, the element of a monster is important because it does extra damage against an enemy monster possessing the opposing elemental traits, such as Fire --> Ice, Ice --> Wind, Wind --> Fire, Thunder --> Water, Water --> Soil, Soil --> Thunder.
The 13 elements are as follows:
Obviously, since these don't all appear in every game, some of this is just speculation. And when combined, usually the new monster takes on the element of the sub-type, so there can be all kinds of changes and disputes. But this way, there are almost 7 of every element.