Class, Sub-class & Multi-class

Actors on the Stage
White Box introduced the world to a game of fantastic adventure and role-playing of three classes. Fighting men, Magic Users, and Clerics, the latter maybe seeming a bit odd, except that it fits in well with the implied milieu - a quasi European medieval setting. The Cleric also brings some unique abilities verses the undead - those popular denizens of tombs and dungeons - and gives the game a class between Fighting Man and Magic User with some abilities of each. The Fighting Man and Magic User are arch-types of the source fiction. Many, if not all,of the heroes of fantastic fiction can be classed as a Fighting Man or a Magic User based on whether their peculiar strength seems to be combat or magical knowledge. Some even combine both.
The character class is one of the genius concepts in White Box. The class instantly tells you something important about the character. The distinct abilities of each class is a way to ensure that no one PC has all the best answers to every challenge and that each player will get a chance to shine in play by using the abilities of their character. It sets the stage for cooperative play and the adventuring "party" of more-than-one. The character class also helps the new player have some idea how to play the character. For me White Box gets it right with just enough difference between classes to make them distinct without over defining the character. For some this lack of further definition makes each member of a class functionally the same, but I feel it is an invitation to define the character in imaginative ways.  Every aspect of the PC not defined by their class is open for the player to define, through description, backstory, emergent play or any combination. Not only does the PC change through leveling, but may change in all the other ways not defined by the rules as a result of play.
Supplement I Greyhawk introduces a fourth character class, the Thief, and in doing so redefines the game as played in the Greyhawk campaign. Character classes define who the story in the game is going to be about, who the major players are to be. White Box is a game about Fighting Men, Magic Users and Clerics. Greyhawk now includes Thieves as the subject of play. The adventures of a thief can now be the story. Inclusion of the Thief character class also poses certain game play questions such as "can the other classes do thief things like open locks or disarm traps or attempt to move silently?"
Greyhawk also gives us the Paladin, a type of Lawful fighter which is commonly referred to as a subclass, but I rather prefer to think of as the first prestige class. Supplement I indicates that a Fighting Man with a Charisma score of at least 17 may achieve paladin status "IF THEY ARE LAWFUL from the commencement of play with that character". I see this behavior qualification as a per-requisite situation more akin to later prestige classes than the more typical beginning ability score requirements of sub-classes. Supplement I doesn't specify at what point the PC gains the paladin status, but I like to see that come out of play and may even be a first level event if that is the agreement of referee and player. The Paladin class adds considerably to the setting Greyhawk and to me helps define the unique milieu of the campaign.
Best of the Dragon presents reprints of three other early sub-classes, the Illusionist, the Bard and the Ranger. All three would see inclusion in the Advanced Game, although the Bard is still "supplemental" or optional, if you will. One of the goals with any new class is to help define the milieu by establishing who the actors in the campaign will be. Another goal is to keep the classes distinct and balanced so that each player feels an equal part of the team. The referee considering a new class may want to inquire how the class will add to the milieu, whether it fits into the desired cast of actors in a beneficial way or whether the class is perhaps not a good fit. As college kids learning to play White Box I don't recall being this deliberate about introducing new classes, or anything new into the game for that matter. We just smashed it all together and made it work, even though it often had unforeseen consequences on our play.
The Multi-class character is one who belongs to two or even three classes. In the Advanced Game it is an option open to all the non-human PCs, but not to humans. I believe this practice greatly alters the milieu and the way the game works. Multi-class characters can obviously fill multiple roles within the adventuring party, although they advance slowly and have many class level limits. They make more sense to me in solo adventures and very small PC parties.
Creating a "custom" character class is quite fun and one of the best ways to reward certain players who are consistently good role-players. The easiest way to do this is by gently altering an existing class. This generally makes the class a bit stronger by adding new abilities, so increasing the experience needed to level is often necessary. When designing a class for my campaign, I try to avoid stepping on the abilities of another class, rather tying the new class into an organization that operates in the campaign and bestows as many social benefits as mechanical ones.
Over the years I have had many players bring an idea to me as referee for a new class usually based on a character from a book they have read. I generally try to make convince the player they can do what they want using an existing class with some backstory and role-playing. Often this together with maybe offering them a single "signature" ability of the fictional character is enough to inspire them to play the character idea.