Unless you're focused on raising Pokemon for use in the Battle facilities, natures don't really matter much to the average player.
In a nutshell, Nature tells how a Pokemon will grow stat-wise. It's easy to tell on the status screen: If there's red text over a stat, that means that stat will grow higher than normal. If there's blue text over a stat, then that stat will grow more slowly.
So for instance, say that I have two Chandelure: One has a Modest nature, and the other has an Adamant nature. The one with Modest will have priority on Special Attack, while slowing down regular Attack's growth, which is very good to have on a Pokemon like Chandelure. However, Adamant will speed up Attack's growth while slowing down Special Attack, which isn't as good.
Then there are the natures that don't really have any effect, like Hardy. They don't affect stats at all. But like I said, unless you plan on doing some serious work at the Battle facilities, you don't need to worry about Natures too much.
