Boots of Travel, Satanic, Linken's Sphere, Manta Style, Monkey King Bar, Butterfly
Boots of Travel is the fastest set of boots and also gives teleportation ability. It's imperative that you are faster than your opponents if you want to hit them with the full potential of your ultimate. Also helps in running and getting to ganks in time.
Satanic is another critical item for Luna. Her starting HP and HP gains through levelups are pathetic, so this item not only provides a boost of +475 max HP (through 25 strength), it also gives her 25% lifesteal per attack, a means of regenerating HP for her.
Linken's Sphere is a defensive item. It's not completely necessary, but you'll find it comes in handy in almost all games you play, because the other team is bound to have a few single target disables of some kind or another, whether they are stuns, hexes, etc. Besides the spell negation every 20 seconds, it gives Luna stats and regeneration for both HP and mana. All around useful item.
Manta Style focuses more on movement speed and damage potential. As an agility hero, Luna can be a pretty beast carry, and it makes sense that the more powerful a hero's regular attack is, the more powerful illusions summoned by them would be.
Monkey King Bar is a main damage item chosen for three reasons. First and foremost, true strike is an incredible ability that makes you never miss. I don't care that Divine Rapier gives +250 damage, you really only deal 2/3 of that over time if you're going up against a hero with a Butterfly or passive evasion. Second, there's a 35% chance of dealing 100 bonus damage and a ministun per attack. Nice damage bonus, but the ministun is where it's at - cancelling channeling spells such as heroes trying to run away by teleporting or Witch Doctor's ultimate. Finally, it gives 80 damage and 15 attack speed.
Butterfly is obvious on any agility hero. +60 damage, +60 attack speed, +4 armor, and 30% evasion.