I wanted to do one sort of magic for my elf archer. In rp, she loves to spend time in the woods with animals and such. What sort of magic describes ''nature'' the most?

Alteration.
That's what I use for my pagan-like warrior.

most of the illusion spells

I was thinking about alteration, only because the icon is a tree though. So it screams for ''nature'' if you ask me.
And magical protection could be handy.

I would agree with Alteration,at least for the oak armor spell.
Illusion possibly but that sounds more like a mage than a forrest loveing wood elf.

Conjuring Familiars fits a little, Alteration is the most apparent for Nature with spells like Oakflesh. If you dont mind killing dragons get Kyne's Peace and Animal Allegiance are very Forest-friendly shouts

http://skyrim.nexusmods.com/downloads/file.php?id=10733. Combined with http://skyrim.nexusmods.com/downloads/file.php?id=10860 of course.

Oakflesh, Stoneflesh, Ironflesh, Ebonyflesh, Water Breathing, Dragonhide, Conjure Flame Atronach, Flaming Familiar, Conjure Frost Atronach, Conjure Storm Atronach, Flame Thrall, Frost Thrall, Storm Thrall, Flames, Frostbite, Sparks, Fire Rune, Fire Bolt, Frost Rune, Ice Spike, Lightning Bolt, Lightning Runem Chain Lightning, Fireball, Flame Cloak, Frost Cloak, Ice Storm, Lightning Cloak, Icy Spear, Incinerate, Thunderbolt, Wall of Flames, Wall of Frost, Wall of Storms, Firestorm, Blizzard, Lightning Storm.

All of those are related to nature or the elements. In my head I tend to split spells into categories of Nature/Earth, Air (lightning, storms), Fire, Water/Ice, Life/Death and Other/Mystical (clairvoyance etc).

For a Shaman I would perhaps use Nature, Air, Frost and Fire. For a forest loving elf who use magic, probably just Nature, or maybe Nature/Air.