Jobs will become available throughout the game. When in a Job, a character will gain Ability Points, or ABP. By gaining ABP, the character unlocks an ability learned from that Job, but only for him- or herself.
Setting a Job will change the stats of the character. What is absolutely vital to understand is the concept that leveling does not in any way raise stats or enhance compatibility. A level 50 Bartz raised as a Monk all the time will make as good a Black Mage as one raised as a Black Mage all that time. HP is changed along with a different Stamina value (since Stamina governs HP) and MP is changed along with a different Magic (Power) value (since Magic governs MP).
Unlike Freelancers, Jobs will have a limited equipment pool. A Knight can equip Swords and Knightsword, but not Harps. A White Mage can equip Robes, but not Heavy Armor.
Jobs can have inherent abilities that are set no matter what. A Monk will always have Barehanded and Counter, a Mystic Knight will always have Magic Shell.
Mastering a Job will unlock the stat boosts (but not penalties!) and most inherent abilities of that Job (if any) for the Freelancer and Mime Jobs. When Bartz masters the Monk Job, he will have the Monk's Strength, Agility and Stamina as a Freelancer, but not the Monk's abysmal Magic Power. In addition, he will gain Counter and Barehanded as inherent abilities, but not the HP boosting support abilities.
Some action and support abilities not only allow a character to execute a feat it would not normally be able to; some also raise stats to pass along the stat of the Job the ability came from. The Knight's Equip Sword support ability also gives the character the Knight's Strength (but only if it were higher than the original Job's Strength). Abilities that grant a spell list often increase Magic Power while equipment options often increase Strength and Agility.
All that said, below you can find all these jobs listed out for you, with all their abilities, bonuses, and equipment. It's more colorful than the text above.
"Musicians whose dulcet tones soothe savage beasts...or kill them."
Duck out of sight to avoid attacks.
Notes: Runs away from the fight, becoming untargetable by any attack. While hiding, no other action can be taken but to eventually return to the battlefield with the !Reveal ability.
Gain the ability to equip harps.
Notes: Gives the character base Freelancer Agility + 11 and base Freelancer Magic Power + 11 and allows the character to equip Harps.
Perform magical chants with different effects.
Notes: Allows the character to sing all songs learned by the party up to that point.
Those stats! So painful. The Bard must have the blues. It has the worst Magic Power of any Robe Job in the game. And the lowest Stamina of any Job. The Bard takes hits like a dusty old marshmallow and has horrible equipment options. Harps are all useless except for Apollo's Harp, and even then only against Dragons and the Undead. The Bard has got to be the worst Job in the game, the !Sing skillset not counted. Offensively completely inept and unsuited for traditional means of gaining offense (since no Rods and horrible survival rates in the Front Row), the Bard should only be used for learning !Sing in areas where Undead are plentiful (thus mastering the Job) and subsequently discarded.
!Hide can be abused by having Zombie characters on the battlefield. !Hide your remaining character(s) and have your invincible Zombies clear the field. Since they'll only attack allies, you'll need Reflect Rings on them and weapons that randomly cast spells, such as the Blitz Whip or Death Sickle. The spells will be re-directed to the enemy, and you win! Beyond that, it's really just a gimmick command. I find Equip Harps is actually useful since the Apollo's Harp can deal peerless damage versus Undead and Dragon bosses, as well as those you target with a Dragon Kiss. Since the Apollo's Harp 'attack' is actually a spell, a Harp-wielding Summoner will be much more proficient with it than a Bard ever will. !Sing is just a great debuff command. I do not much care for the stat-increase Songs, but Requiem and the Stop/Confuse songs (Romeo's Ballad and Alluring Air) are very useful in large stretches of the game. !Sing is great on everything.
No stats to speak of, no inherent abilities. Nada. In Final Fantasy Tactics, only men can be Bards, by the way. Female Bards are an abomination.