Is there a list of effects that you can enchant items with? If not, then is it possible to enchant an item with constant effect spell absorption?

Also, do you actually have to have the type of spell to put that effect on an item, like in Oblivion

Any spell effect can be enchanted into any piece of armor or clothing or any weapon. Armor, clothing, and weapons can all use cast-when-use enchantments, weapons can use cast-when-strike enchantments (though this does not work properly with bows), and everything can use constant effect enchantments if the soul being used is 400 points or more. Golden Saints are the best source of 400 point souls but they don't appear until you get to the higher levels.

So, yes, you can make an item with constant effect spell absorption.

But do I need the specific spell? For example, do I need a Restore Health spell before I can Enchant an item and give it a Restore Health effect?

Yes, you need to have a spell in order to use it to make a custom spell or an enchantment. Not all spells effects can be made into enchantments; most can but not all. You can look in the CS under magic effects to see if the enchantment box has been checked. For example, restore magicka cannot be enchanted or absorb magicka even though there is an enchanted item in the the game with that effect.