STR (Strength)
primary attribute for increasing melee ATK
AGI (Agility)
sole attribute for improving speed of all normal melee, range, and magic attacks
sole attribute for increasing block rate
sole attribute for increasing evasion rate
secondary attribute for increasing melee and range ATK
CON (Constitution)
sole attribute for increasing base hitpoints
sole attribute for increasing resistance to knockdown from normal attacks
DEX (Dexterity)
primary attribute for increasing range ATK
sole attribute for increasing critical rate
sole attribute for improving accuracy
sole attribute for improving efficiency of traps and constructions (turrets, blades, etc)
shared attribute (with INT) for improving efficiency of healing skills
INT (Intelligence)
sole attribute for increasing magic ATK
sole attribute for increasing magic RES
shared attribute (with DEX) for improving efficiency of healing skills
CHA (Charisma)
sole attribute for improving efficiency of summonables (Catherine's dolls, Wicky's golems and pets, ambers, spinelles, etc)
Footnote
Every point in STR above 50 improves melee ATK by 2% and every point in STR under 50 reduces melee ATK by 2%. Likewise for DEX and ranged attacks. Every point in AGI above 50 improves both melee and ranged ATK by 1%. There is no penalty for AGI under 50. This is applicable to all melee and ranged attacks, regardless of weapon types. The combined effects of STR/DEX and AGI have been revised several times. Currently, in sGE at least, the combined effect is additive, and the crude formula for effective melee ATK for AGI above 50 is
effective ATK = base ATK * [1 + (STR-50)/50 + (AGI-50)/100]
and for AGI under 5 0
effective ATK = base ATK * [1 + (STR-50)/50]
Substitute STR for DEX for ranged ATK.
Hybrid melee-and-range stances, i.e. Heaven or Hell and Avance Garde, are affected by STR, DEX, and AGI differently. The formula for the displayed effective ATK of hybrid melee and range stances, is
effective ATK = base atk * [1 + (STR-50)/50 + (AGI-50)/100] * [1 + (DEX-50)/50 + (AGI-50)/100]
What this means for the hybrid stances is that adding points to either STR or DEX gives the same ATK increment, and that adding points to AGI adds more to ATK than either STR or DEX does. Both of these are true if observed solely from the character stat panel, but the actual performance of these hybrid stances depends on the attack and skill animations. In practice, for both Heaven or Hell and Avance Garde, STR is slightly more effective in raising normal attack damage, DEX is slightly more effective in raising hybrid skill damage, and AGI is a balance of both. (As for which stat point to add for hybrid specialists, the answer remains subjective and dependent on the individual's play style.)
DEX improves accuracy of melee and range stances by (DEX-50)*0.5, i.e. every point in DEX above 50 improves accuracy by 0.5 and every point in DEX under 50 reduces accuracy by 0.5. Melee stances have 100 base accuracy unless stated otherwise, and characters with under 50 DEX will have reduced accuracy for those stances.
DEX improves base stance critical rate at a factor of DEX/50. The formula for effective critical is 5 + (DEX/50)*(stance bonus) + (additional crit from equipment options).
Every point in AGI above 50 speeds normal attacks by 1% and every point in AGI under 50 slows normal attacks by 1%. AGI does not affect skill animation, skill casting, or character movement speed. STR does not affect heavy weapon attack speed.
AGI improves base Block at a factor of (AGI+50)/100. Effective Block is therefore [(AGI+50)/100]*(stance bonus + base shield block) + (additional block from equipment options). Effective Evasion is calculated in a similar way.
INT improves base RES from equipment and stances at a factor of (INT+50)/100. Characters with INT under 50 are penalize accordingly. (Note: the basic RES bonus listed on this site is currently incomplete/inaccurate for some stances, and base armour RES, like base Block, is a hidden attribute not available in-game.)
DEX and INT contribute equal weightage in the calculations for heal amounts of all healing skills.
Knockdown rates from skills are fixed and unaffected by STR and CON.
Credits to Ginjiwai for additional assistance and information on various entries, including, but not limited to, formulas for RES, Block, Crit, and STR.