Destiny’s story and lore is one of the least explained and least known things about the game so far. We got a taste of the story in the beta, with two major storylines having started: The Hive’s presence on Earth and the story of Rasputin, the last Warmind.
Keep in mind this is only a small sliver of the lore!
To get invested into more Destiny lore, be sure to check out your Grimoire cards.