What is the purpose and functionality of the supertypes and subtypes of Magic: The Gathering?

Having Supertypes means there are card classifications that can apply across multiple card types. So even though Artifacts and Lands are very different types of card, both kinds can be 'Legendary' and affected the same way by the Legend rule (all supertypes have special rules to go with them).

On the other hand, subtypes are used to create differences between cards of the same type. It means a Creature - Elf Shaman, and a Creature - Goblin Warrior, are two different things. They are both Creatures and so affected by things that affect creatures, but special effects that work on Elf cards will work on the former and ones that work on Goblins will affect the latter.