Natures Explained
Natures change a Pokémon's stat growth by boosting one stat by 10% and lowering another by 10%. Some natures are neutral and don't change stats (technically, they increase and decrease the same stat).
What a nature changes
A nature affects only stats (Attack, Defense, Sp. Atk, Sp. Def, Speed). It does not change HP.
Competitive rule of thumb
Most special attackers prefer a nature that boosts Special Attack or Speed and lowers Attack. Most physical attackers prefer a nature that boosts Attack or Speed and lowers Special Attack. List of Natures
FAQ
Do natures matter in-game?
Yes, but mostly at higher levels or when you care about optimized stats.
Can a nature be changed?
In modern games, mints can effectively change the stat effects of a nature.