Once you've chosen your realm you will be taken to a kind of blank screen (blank in the sense that there are no characters there). On the right hand side near the bottom will be a button that says "Create Character". You'll want to click on that.
This will open up the character creation screen.
On the left hand side you choose your faction (Alliance or Horde), race, and gender. On the right hand side you choose your class. Each race has its own class combination and no race can be every single class. I will go over the races and classes in a bit.
The Back and Next buttons at the bottom move on to customizing a character. You can customize things like skin color, face, hair, etc. Each race also has personalized customizations. Some races allow for ear/jewelry customizations, others allow for horn customization and so on.
Races and Classes
Now lets cover races and class combinations.
Alliance
Human - Priest, Rogue, Warrior, Mage, Hunter, Warlock, Paladin, Death Knight, and Monk
Dwarf - Priest, Rogue, Warrior, Mage, Hunter, Warlock, Shaman, Paladin, Death Knight, and Monk
Gnome - Priest, Rogue, Warrior, Mage, Warlock, Death Knight, and Monk
Night Elf - Priest, Rogue, Warrior, Mage, Druid, Hunter, Death Knight, and Monk
Draenei - Priest, Warrior, Mage, Hunter, Shaman, Paladin, Death Knight, and Monk
Worgen - Priest, Rogue, Warrior, Mage, Druid, Hunter, Warlock, and Death Knight.
----
Horde
Orc - Rogue, Warrior, Mage, Hunter, Warlock, Shaman, Death Knight, and Monk
Forsaken - Priest, Rogue, Warrior, Mage, Hunter, Warlock, Death Knight, and Monk
Tauren - Priest, Warrior, Druid, Hunter, Shaman, Paladin, Death Knight, and Monk
Troll - Priest, Rogue, Warrior, Mage, Druid, Hunter, warlock, Shaman, Death Knight, and Monk
Blood Elf - Priest, Rogue, Warrior, Mage, Hunter, Warlock, Paladin, Death Knight, and Monk
Goblin - Priest, Rogue, Warrior, Mage, Hunter, warlock, Shaman, and Death Knight
----
Neutral
Pandaren - Priest, Rogue, Warrior, Mage, Hunter, Shaman, and Monk
Once you have your race and class picked out and have them customized the way you want them, just keep hitting the next button and you'll need to give your character a name. Choose a name fitting for the race (there are plenty of WoW naming sites around). Make sure its nothing vulgar, and hit "Enter World". If the name is already taken, you'll be given an error and have to choose a new name.
Break Down of Classes
Priest - Basic healing class. Wears cloth armor only. Can also be used for dealing damage depending on how you set them up. Can also revive fallen allies.
Rogue - Damage dealing(DPS) melee class. Relies on poisons and sneak attacks. Can wear cloth or leather.
Warrior - Melee DPS and tanking class. Able to dish out and withstand high amounts of damage. Wears cloth, leather, mail, and plate.
Mage - Basic caster with the ability to use fire, frost, or arcane magic. Uses AoE (Area of Effect) spells to target multiple enemies at once. Can also use Polymorph to temporarily turn enemies into harmless critters. Wears cloth only.
Druid - A hybrid class that can be a healer, DPS, or tank depending on their set up. Has the ability to shapeshift into animal forms such as a cat or bear, even a bird to allow you to fly. Wears cloth and leather.
(Druid forms are; Bear (tanking), Cat (melee DPS), Moonkin (ranged DPS), Tree (healing), Cheetah (increase running speed), Seal (increase swim speed), and Bird (allows you to fly).)
Hunter - Ranged DPS that uses guns and bows. Able to tame animals as pets to fight along side them. Pets are permanent unless you release them yourself and they will level up and gain abilities with you. Wears cloth and leather to start out then gets mail at like lv40 or so.
(Hunters are able to tame any creature listed as a beast if its their level or lower. So a lv50 hunter could tame a beast that was anywhere between lv1-50, but couldn't tame one that was higher level than them. Low level pets will automatically scale up in level once they're tamed.)
Warlock - Caster class, much like a mage. Has the ability to summon demon minions to fight for them. Wears cloth only.
Paladin - A hybrid class thats like a combination of priest and warrior. They're basically holy warriors than can heal, tank, or DPS. They can wear cloth, leather, mail, or plate.
Shaman - Another hybrid class that can play healing or dps. They use totems to heal or do damage. Can turn into a spirit wolf. Wears cloth, leather, and mail.
Death Knight - An undead warrior. Death Knights were former agents of the Scourge that broke free from the Lich King and rejoined their respective factions. Death Knights are the only class that does not start at lv1 and you have to already have a lv55 character on that realm in order to play one. They can tank or DPS and wear cloth, leather, mail, or plate.
Monk - Not real clear on the monk class as I've never played Mists of Pandaria. I think they're a hybrid class. At the very least they can heal or DPS. They use chi to power their attacks and spells. They wear cloth and leather I think.