Since there is a a lot of confusion about how the class system works in FFXIV: ARR, I decided to write this guide to clear up matters. I need to make a bunch of corrections and edit this some more, but this is what I have so far. If you see any mistakes or have anything to add, please let me know.

First and foremost, you must understand the difference between disciples, classes and jobs else get confused.
Disciples :
There are 4 disciple types in the game. Disciple types in FFXIV are nothing more than general headings for the 4 different methods to do anything in game. They are Disciples of War (using physical weapons for battle), Disciples of Magic (using magic for battle), Disciples of the Land (gathering), and Disciples of the Hand (crafting). Disciples have no levels and cannot progress in any way, they are nothing more than descriptive headings.

Classes:
Each class can be leveled from 1 to 50, but unlike many other games, a single character can level up as many classes as they wish. To change your class, simply equip the weapon / tool specific to the class you want to become anytime you are NOT in combat. You must finish the initial tutorial quest line before you can change jobs (the level 10 tutorial quest allows you to change class but it is not practical to collect all the classes until you unlock airship travel from the level 15 story quest). You must also talk to a different NPC once for each class to unlock it (the NPC is always the head of NPC guild for the class found in one of the 3 capital cities). If you level a Marauder to 35 and then decide to try healing as a Conjurer, just change your weapon to a wand and you will be set back to level 1 to start leveling as the new class. If you change back to a Marauder, you go back to level 35 and continue leveling it from where you left off. Your base stats change depending on your current class and level.

Each disciple type has classes inside of them and these are what you will be leveling for the entire game. Disciple of War's classes are Archer (bow, ranged dps), Gladiator (sword, melee tank), Lancer (spear, mid-ranged melee DPS), Marauder (Two-handed axe, heavy melee DPS), and Pugilist (Claws and knuckles, avoidance/elemental melee DPS). Disciples of Magic's classes are Arcanist (Books, pet based offensive spellcasting with healing if scholar job is used), Conjurer (Wands, support and offensive spellcasting), and Thaumaturge (Staves, debuffs and offensive spellcasting). These classes are what you will be using when fighting enemies. They level up as can be expected of any other MMO, through killing, FATEs (field events) and completing quests (which mostly involve killing stuff).

Disciples of the Hand (crafters) and Disciples of the Land (gatherers) also have their own classes so they need to level from 1-50 just as any Disciple of War or Disciple of Magic; however, they do not gain their experience through battle, but rather, through crafting, gathering, and special non-combat quests. A disciple of the hand could easily level from 1-50 without ever touching an enemy simply through crafting and it's associated quests along with the use of the in game auction house while a Disciple of the Land gains access to abilities that let them stealth past enemies so that they can gather resources from anywhere even if they are completely incapable of fighting. Disciples of the Hand's classes are Alchemist (Alembics), Armorer (Raising Hammers), Blacksmith (Cross-pein Hammers), Carpenter (Handsaws), Cullinarian (skillets), Goldsmith (chaser hammers), Leatherworker (Headknives) and Weaver (Needles). Disciples of the Land's classes are Botanist (hatchets), Fisher (Fishing rods) and Miner (pickaxes).

Cross Class Skills
In every individual class at level 5 and every 5 level after that, you unlock one slot where you can put in an ability from another class that you have unlocked through leveling it. You never have to use cross class slots for abilities from your current class; you always have access to all of them. At max level (50), you will have 10 cross class slots. This gives you a lot of flexibility and encourages you to try out other classes.

There are some restrictions however. Many, but not all, usable skills earned in the first 34 levels of other classes can be slotted; however, most skills earned above level 34 are class exclusive (cannot be used in cross-class slots). Many skills are also restricted by their discipline type (for example, all offensive spells require that you are a Disciple of Magic, and most Disciple of the Hand and Land skills cannot be used with Disciples of Magic or War). Other restrictions may also apply such as requiring a shield or a melee weapon. Cross-class skills scale in power based on the level of the class it is equipped on.

Skills used in cross-class slots are just as effective as when they are on their original class; however, they may not perform just as well because of two main reasons. First, when an ability is used cross-class, there is a good chance that the ability is not based off the same stats as the class it came from has an abundance of. For example, the cure spell uses the MND (mind) stat, a stat rarely used by any non-healing class thus using cure on any Discipline of War character will likely result in poor healing. Second, every class also has access to passive 'traits' which cannot be used cross-class and often enhance the effects of spells. For example, Lancers have an ability called 'invigorate' which instantly replenishes 200 TP (the resource used for most physical attack abilities). While any class can equip this ability with equal effect, at level 36 only Lancers gain access to a passive trait called 'exhilarate' which increases the TP gain of 'invigorate' from 200 to 450.

Jobs
Jobs are very simply the sacrifice of versatility in exchange for specialization. A lot of players prefer to use the specialization of jobs when in parties while the versatility of classes is often better suited to soloing. Every Disciple of War and Magic class in the game has access to one or more jobs.

Jobs share the level and ALL abilities and passive traits with the class it is based on. Gladiators for example may specialize in tanking and become Paladins by equipping a Paladin soul crystal. Paladins have access to all Gladiator abilities and passive traits plus a few very powerful Paladin-exclusive tanking abilities that can only be earned through quests. The trade-off is that they only get half the cross-class slots; one every 10 levels instead of the usual one ever 5 levels (so a level 45 Paladin gets only 4 slots instead of the usual 9) and they may only use cross-class skills from two predetermined classes (in the case of a Paladin, Conjurer and Marauder). Remember, Gladiator is NOT cross-class for a Paladin, a Paladin IS a Gladiator that is specialized in tanking.

The quest to unlock a Job always requires that the player is level 30 plus 15 levels in one other predetermined class (in the case of Paladins, 15 levels of Conjurer is also required). While a class and its associated jobs share level, each has its own set of predetermined stats (so a Paladin does not have the same stats as a Gladiator despite having the same skills, traits and level). A player may un-equip a soul crystal at any time to return to their base class, thus regaining their cross class slots and having access to more cross-class skills but they lose access to their specialized Job-only skills. It should also be noted that some equipment is restricted to certain jobs.

To add to clarity regarding levels, whenever you are leveling a base class (like Gladiator), you are also leveling all its jobs to the same level at the same time (so your Paladin job will always be the same level as your Gladiator class). The inverse is also true. If you are leveling a job, you are leveling all other jobs of the same base class (even the ones you don't have or have not been released yet) and the actual base class at the same time. As another example, the Arcanist class has two jobs, Scholar and Summoner. Your Archanist, Scholar and Summoner levels will always be the same, even if you don't have Summoner yet. A summoner simply adds more specialized offense summons to the Archanist while the Scholar adds more specialized hots and support.

The following is a list of the current jobs. Text in bold indicates the base class, italics indicate the available cross class skills and (Lv. X) indicates what level is needed to unlock the quest to obtain the job.
Bard: Archer (Lv. 30), Pugilist (Lv. 15), Lancer
Paladin: Gladiator (Lv. 30), Conjurer (Lv. 15), Marauder
Dragoon: Lancer (Lv. 30), Marauder (Lv. 15), Archer
Warrior: Marauder (Lv. 30), Gladiator (Lv. 15), Pugilist
Monk: Pugilist (Lv. 30), Lancer (Lv. 15), Gladiator
Summoner: Arcanist (Lv. 30), Thaumaturge (Lv. 15), Conjurer
Scholar: Arcanist (Lv. 30), Conjurer (Lv. 15), Thaumaturge
White Mage: Conjurer (Lv. 30), Arcanist (Lv. 15), Thaumaturge
Black Mage: Thaumaturge (Lv. 30), Archer (Lv. 15), Arcanist