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SWORD
This is the main offense class. If you're making a character a fighter character this is pretty important. Skills you gain in this Category include stuff like まほうけん (Magic Sword) and なぎはらう (Nagiharou) which are skills which will appear in your combat menu in a battle if equipped.
Magic Sword is the base skill for the Sword category so when you're in that category you don't have to equip it. Basically for any attack spell that you already have, you can use the Magic Sword attack for. Think of it as an elemental aided physical attack. Just like with spells, you can use any Magic Sword when you have the Sword category equipped, but when you are equipping the skill while using another category, you can only use the Magic Sword attacks up to whatever level of the Magic Sword skill you have gained so far.
Nagiharou is a physical attack that hits all enemies in a horizontal row, but with less damage than a normal attack. As the level increases for this, it does more damage and becomes closer to being able to replace your normal attack completely.
Other skills in SWORD are HP吸収 (HP absorb) which gives you a percentage of damage you deal with a normal attack, クリティカルダメージUP (Critical Damage UP) which increases the damage multiplier for a critical hit, and MP吸収 (MP absorb) which does the same thing as HP absorb but for MP instead.
WHITE
This is pretty much the white mage job from Final Fantasy games. As you can see, Sakaguchi isn't very creative with naming. ホワイトマジック (White Magic) is the base skill here, and also has levels. The levels once again only matter if you're equipping it on another category, because when you are in this category you have access to any White Magic spells you have found or bought.
Useful spells in this category include stuff like アイテム効果アップ (Item effectiveness up)and 連続アイテム (Concurrent item use). The first one increases the effectiveness of items when used by the character with the skill equipped, and the second one allows the character to use two items in a single turn. For a healer, skills like these are really helpful.
When you're high enough in rank you will also learn ホワイトジェネレート (White Generate) which, when equipped will regenerate the HP of that character as you move on fields. This ONLY works on fields and dungeons, so if you're in a town you still have to use an inn. Don't be a cheapass.
BLACK
This is the black mage job! Woo hoo! ブラックマジック (Black Magic) is the black equavilant of White Magic and works the same way so I won't repeat. This is the base skill for BLACK.
Other skills in this category include ブラックジェネレート (Black Generate) which regenerates MP instead of HP, and is helpful because an offensive mage will usually use up much more MP. Later on you also get a skill called 最大MP+25% (MAX MP +25%) which is pretty self explainatory... I hope!
BARRIER
Barrier is one of the two support magic categories in the game. バリアマジック (Barrier Magic) is the base skill, and works like all other sorts of magic class skills. Barrier magic in general will be stuff like shield and shell. Stuff that create barriers to protect you in battle.
The other skill to look out for is バリバリア (Bari Barrier) which is a field skill. When a character has a field skill equipped, you can set up to two field skills on your LB and RB buttons in the field menu (RT). The Bari Barrier creates a shield around your character on the field and will destroy weaker enemies instantly and still net you EXP and SP. Be warned though, this still takes up MP - it's not a free skill. There is a number after Bari Barrier to indicate how strong it is, you get stronger versions as you rank up.
Another skill you'll get is 魔法の壁 (Magic Wall) which is a Statistic skill. This specific skill ensures your Magic Defense statistic is the same as when you have the Barrier category equipped. Because the base magic defense stat in the Barrier category is higher than in other categories, this skill will allow you to keep the buff if you wish while in another category.
MONK
This is very similar to the Monk job in Final Fantasy games. The base skill in this category is チャージアタック (Charge Attack) which allows you to charge your normal attack just like you can charge spells. The longer the charge, the more damage you deal. It's really quite powerful.
Other skills include the field skill コイオーラ (Koi Aura) which when used makes enemies chase you, カウンター (Counter) which gives the character a % chance to do a counterattack when hit, and ヨガ (Yoga) which is an active skill in battle which allows the character to recover HP. Yoga is also a chargable skill.
POWER
Power is the other support magic category. パワーマジック (Power Magic) is the base skill, and covers mostly status spells. Things like Quick, Slow, Frog, Sleep, etc all fall under Power Magic. Another skill in this category is 効果時間+50% (Time Efficency +50%) which allows the character to pull off spells faster.
ARMOR
Armor is actually quite an original category even though the skills are more or less some of the stuff that the Monk could do in FF games too. Basically in this game the Monk category is much more offensive, with the defensive elements split into Armor.
This category has skills like かばう (Cover) where the character will have a % chance of protecting an ally when attacked, マジックディフェンス (Magic Defense) which raises your magic defense when guarding, 最大HP+25% (MAX HP +25%) and 戦士の壁 (Warrior's Wall) which is a statistic skill that matches the defense stat with the Armor category.
I just realized I have no idea what the base skill of Armor is.... shows how much I've used it... haha.
ASSASSIN
This is probably my favorite category. It's pretty much the Thief job in FF, but more fun! The base skill is セキュリティ (Security) which prevents enemies from stealing from that character. Skills you get early on are ぬすむ (Steal) and シビレダマ (Numb bomb). Steal is an active skill in battles, you have to select it. Numb bomb is a field skill that throws a bomb which paralyzes enemies affected by the blast on the field. This makes it really easy to get back attacks or to evade enemies.
Other interesting skills include 交渉術 (Bargain Skill) which gets you better prices in shops when equipped, 忍者のこなし (Ninja Vessal) which matches your speed stat to that of the Assassin category and ステルス (Steath) which is a field skill that makes you invisible.
Later on you will also gain skills like ぶんどる (Plunder) whereby you will automatically attempt to steal when doing a normal attack, and 遠距離攻撃 (Range Attack) which allows you to freely target enemies in the back row as well with any physical attack.
COMBINATION
This is a special category which is VERY important. The base skill is アクセサリ+1 (Accessory +1) which allows you to equip special equipment on an extra slot. These are equipment which would otherwise be unequippable because they do not fall into the "Bracelet", "Ring", "Necklace" or "Earring" equipment sections.
The next important skill this category provides is スキル+2 (Skill +2) which is actually skill +1, since you have to equip this skill and it gives you 2 more. In reality you're only getting one more skill. False advertising!
As you rank up this category, both Accessory+1 and Skill+2 will increase into Accessory+2 and Skill+3, Skill+4, etc. This makes the category very worthwhile exploring even though it doesn't actually have any real skills of its own.