The reason it sounds like a lot of nonsense when listening to melee chant about the numbers is because that's mostly what it is. The stats really do have an effect, but ultimately, it's damn small on the scale of things. In FFXI most of your effectiveness is determined by your level, not your gear. A character in average gear is virtually identical to one in the very best gear imaginable -- maybe 10-20% difference. It might be easier to do things with 18 players all in top gear, but there's nothing those same players couldn't do in regular gear.
People still want to improve their characters after they hit 75 -- gotta be *something* to do besides level a new job or merit forever. So, melee start crunching numbers and struggling over a few percent here and there.
Anyway, here are some additional details on stats if you're really curious. Most of Gabryel's list is accurate; this just adds some minor tidbits and corrections.
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ATK: How effective you are at overcoming your target's DEF (more atk, more damage). Generally rather linear with more atk = more damage. At a rought guess, +10 ATK = +1% damage.
DEF: How effective you are at taking damage (more def, less damage). Like ATK, this one is also pretty linear.
STR: +1 = +0.5 atk. STR determines how effective you are at overcoming your target's VIT and factors into many physical WS's. Has a stepwise effect -- once over the VIT threshold, returns from more STR are not linear wrt. damage.
DEX: +1 = +0.5 acc & 0.01% crit given. For main THF only, +1 DEX = +1 base damage for sneak attack. Acc&Crit are empirical guesses -- no one *knows* for sure, but it's probably pretty close.
AGI: +1 = +0.5 r.acc, +0.5 eva, & -0.01% crit taken. Seems to increase passive defensive skill activation (parry, counter, guard). It does not increase any offensive JT activation rates (eg. double & attack). For main THF only, +1 AGI = +1 base damage for trick attack.
VIT: +1 = +0.5 def and reduces damage from high STR attacks. Stepwise, like STR. Beat your target's STR and you take a lot less all of a sudden.
CHR: Improves charm success & duration (bst, brd) and increases killer-effect activations. It doesn't help Enmity or Provoke. Primary stat for Dancing-Edge WS (weird, but true).
INT: +1 = +0.5 m.atk, m.acc, & m.def for int-based spells (mostly blm'ish things). Primary for all Ninjutsu -- damage and debuffs.
MND: same as int but for mnd-based spells (mostly whm'ish stuff). Determines base damage for monk's Chi-blast.
HNM tidbit -- I've heard 100 STR & 100 VIT touted as magic numbers for either tanking or doing damage to HNMs. On XP, more str/vit doesn't seem to much matter; you're already past critter str/vit for the most part. This kind of matches what I've experienced myself, but the jump difference isn't quite as "omg, wow!" as I've seen on web. It's more "oh, that's better".
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The one thing that's true about all stats is that it takes a whole lot of +stat to actually make a real difference. You usually get much more out of direct +atk, +acc, +def, +etc.. It's also why the Osode doesn't parse well for DD during XP compared to other things. Kinda funny that such a highly desired item by so many isn't really all that hot; it's more of a good general-purpose body that isn't top in any one category.
IMHO, this is a very good thing. I've played EQ and WOW and they're exactly the oposite. In those games, your character doesn't matter much. It's all gear. Your gear in those games sways character performance by 10x or more. I made a level 1 char in EQ for fun with ~1500hp by giving it gear I didn't need. A normal level 1 character has ~10hp. There are things in those games you can't do without sufficiently nice gear.
So, if you think the struggle for gear here is kinda ridiculous -- well, you're right. But, it's all we have left to work on at 75.
