When I first played Ruby (which was the first game with Abilities and Natures) I didn't care about natures, I just used that Pokemon.
But with each generation gone by, I suddenly heard about that large group of competitive players who prefer certain natures.
Pearl, Platinum and SoulSilver I just catch a Pokemon, still not worrying about the nature but more about the abilities.
When I catch a Buneary, I always make sure it has Run Away and not Klutz.
A Pokemon I really started caring about natures was Skuntank, I love this thing but every time I get one by trading with my brother's Diamond or hatching a Stunky, I always ended up with either a neutral nature or one that makes it terrible (I remember getting a Modest Skuntank once) and I only go for Skuntanks that have Aftermath cos I hate Stench.
It was in Black that I really started going for certain natures and abilities more seriously.
When I catch a Minccino, I only take ones with Technician (as Marriland recommends).
When I caught an Elgyem, I was fearing she'd have a terrible nature but I ended up with a Quiet nature, which made her terribly slow but at least a Special Attack boost. I didn't evolve her until the post-game cos of that Speed stat, I still like Elgyem but its Speed is a let-down (no wonder why Gothitelle became my main battling Psychic in Unova).
And when I caught a Venipede, he had an Adamant nature, which is great and now he is the ultimate Scolipede! I couldn't believe he had the right nature.
I also have a Leavanny, with a Lonely nature. Hello high Attack stat!
My Galvantula wasn't that good, Brave nature and Unnerve ability (terrible for Galvantula) so sadly I don't get those good Thunders.
I was inspired by Elesa and caught myself an Emolga, with a Quiet nature! Special Attack power boost, but the Speed lowering didn't do much as Emolga is very fast anyway. Same case with my Swoobat, thankfully he has Unaware and not Klutz (I HATE KLUTZ!)
My Garbodor turned out to be the most terrible Pokemon I ever caught. Bold nature + Weak Armor = totally stinks.
My Reshiram has the right nature, oh yeah, Modest baby!
I caught a Grumpig and Mr. Mime via Poke Transfer, and sadly they are Adamant, sob.
And my Mantine and Lanturn (also via Poke Transfer) were given Special attacks by me only to discover they were Jolly.
Jolly? Oh, Jolly, why why! (sorry, I was imitating Marriland there)
And my Gabite (Poke Transfer caught) sadly has a Calm nature, looks like she'll never be used competitively because of sacrificing some of her Attack for a darn Special Defense boost!
I have a Misdreavus that is a Special Attacker, and yep, cursed by a Jolly nature.
My Probopass is hindered by a stupid Timid nature, sob.
At least my Luxray got the right nature, Naughty baby!
My poor Hariyama is TIMID! Argh!
My Rampardos is female, and a Brave nature, yeah baby!
Fearow, sadly has a Bold nature. Damn!
And my worst-case scenario - I caught a Golett only to discover it was Klutz, I went back to Dragonspiral Tower, caught another one, this time with Iron Fist but wait for it, I got the worst possible nature and that nature is...
And my Stunfisk is Adamant, stupid.
My Shuckle is Modest, terrible as it is.
And my Bibarel is darn TIMID!
At least my Heracross, Gliscor and Absol ended up with good natures (Naughty for Heracross and Adamant for Gliscor and Absol)
I find Graveler terrible, not only that 4x Grass/Water weakness, but mine has a... CALM NATURE! ARGH!
At least my Lapras, Ninjask, Hitmontop and Swellow got the right natures.
I remember catching my Stoutland as a little Lillipup, she hit hard as a Herdier but not really overwhelming, and that's because she is Timid! Timid stinks for Stoutland so I caught one more Lillipup (male this time) and he is an Adamant baby!
And one more worst case scenario, I wanted to train a Pawniard, caught one but sadly she has the worst nature you could get for Pawniard, which is a...
So I had to catch another Pawniard, same gender and ability as before, but a Gentle nature. I raised the Gentle one into a Bisharp eventually.
And with Axew, I only accept ones with Mold Breaker cos Rivalry is dependent on its foes gender. No wonder my Haxorus sucked, I went and caught a Mold Breaker Axew eventually.
For Pidove, Tranquill and Unfezant I only catch ones with Super Luck. Big Pecks is useless.
To test for a Pidove, Tranquill or Unfezant with Super Luck, use a Defense-lowering move like Tail Whip, Screech or Leer. If Defense is lowered it has Super Luck.
Luxray also have Rivalry, but Intimidate is so much better!
So yeah, a lot of my reviews there!