Along with IVs and EVs, Natures are the only variables that you can use to boost the final stats of your Pokemon. Most Natures increase one stat by 10% (or x1.1) and decrease another by 10% (or x0.9). Therefore, advanced trainers look to give their Pokemon the ideal Nature to match the required role of the Pokemon in their team.
A new feature introduced in Emerald for the GBA is the ability to transfer Natures from parent to egg. To achieve this, an Everstone has to be attached to the mother, and there will be a 50% chance that the mother's nature will be transferred. This feature isn't available in Ruby/Sapphire/FireRed/LeafGreen verisons, meaning that you'll have to continuously hatch new Pokemon in the hopes of finding a useful Nature. Nature-transferring is also present in Diamond/Pearl.
The next few sections will completely outline the practicality of every Nature. I have allocated them into groups determined by the Stat they boost.
I small kernel of advice I'd like to throw out there; Try and go for Natures that decrease either Attack or Special Attack. Most Pokemon use either of those Stats if they are going to directly attack a foe, and the other stats (namely Defence, Sp.Def and Speed) are always important for all Pokemon. For example, having a Groudon with a Naughty Nature (+Atk, -Sp.Def) might seem like a good idea initially, but a useful stat is being decreased, which isn't helpful.
I'm sure there are some who use "mixed" Sweepers; that is, Pokemon which use both Physical and Special Attacks in a battle. While the element of surprise might be enough for some, I personally don't think sacrificing precious EVs is worth it. Then again, if you are going to try using a Mixed Sweeper, then you'll probably want to boost your Speed stat with the appropriate Nature.
Please note that this Guide is analysing the Natures. If you want a quick-reference list, then please look up Serebii or the straightforward list in Pokemonline.
Hardy, Docile, Serious, Bashful, Quirky (all +N/A, -N/A)
These Natures don't boost anything at all. Therefore, two Kyogre with exactly the same IVs, EVs and Level will have exactly the same stats as each other, if both have natures from this group. These are the most useless Natures. They don't boost anything, meaning that any redundant stats will not have anything taken from them in favour of another.
Pokemon with any of these Natures must be kept as a last resort. For example, I only kept my Docile-Natured Groudon because it's IVs were exceptionally high. If you have the choice or the time, go for another Nature.
Lonely (+Atk, -Def)
Brave (+Atk, -Spd)
Adamant (+Atk, -Sp.Atk)
Naughty (+Atk, -Sp.Def)
The only useful one here is Adamant; the rest lower stats that are always good for all Pokemon.
Bold (+Def, -Atk)
Impish (+Def, -Sp.Atk)
Lax (+Def, -Sp.Def)
Relaxed (+Def, -Speed)
The first two Natures here are great; depending on the Pokemon you're using, I recommend you choose either Bold or Impish. For example, if you had a Venusaur and wished to give it an edge against Physical Attacks, then Bold would do very nicely, as Venusaur traditionally uses Special Attacks, so no point in lowering that stat with Impish. Lax just makes no sense (sacrificing one Defensive stat for another?) and Relaxed is a viable option if you really aren't bothered about your Pokemon's Speed.
Mild (+Sp.Atk, -Def)
Modest (+Sp.Atk, -Atk)
Quiet (+Sp.Atk, -Speed)
Rash (+Sp.Atk, -Sp.Def)
This practically mirrors the Attack Natures; Modest is the only one worth considering here. The rest just deduct from other Stats that will make a positive difference in a battle.
Calm (+Sp.Def, -Atk)
Careful (+Sp.Def, -Sp.Atk)
Gentle (+Sp.Def, -Def)
Sassy (+Sp.Def, -Speed)
Similar pattern as with the Defence stat. Only the Calm and Careful Natures are the obvious ones. Gentle is one I think is just as useless as the Neutral Natures overall, and Sassy is... For certain Pokemon and certain roles in a team. I'm just covering myself in case somebody finds a new-fangled way of making use of the Sassy Nature.
Hasty (+Speed, -Def)
Jolly (+Speed, -Sp.Atk)
Naive (+Speed, -Sp.Def)
Timid (+Speed, -Atk)
You can't go wrong with the Jolly or Timid Natures. Both of them make fine Natures for Physical and Special Sweepers respectively. The other two are not ideal, as yet again, you're losing stats that are fundemental to all Pokemon.