You gain money in World of Warcraft by completing some of the quests (some of them have no monetary reward associated with them), killing mobs and looting their corpses, selling excess inventory items to the vendor NPCs or on the Auction House, or learning one or more professions. Don't forget to pick a profession and skill it up as much as you can in the early levels. Trying to max out your profession after you hit a very high level (60+) is a tedious and very undesirable process. As you go up in level, both the quest rewards and the money from mobs increase, as a general rule. Also, humanoid and undead mobs drop loot for more cash per kill than any of the beast type mobs who aren't quest-related. Upon completion of a quest at your maximum level (60 during original, unexpanded WoW, or Vanilla WoW, 70 during The Burning Crusade, 80 during Wrath of the Lich King, 85 during Cataclysm, and 90 during Mists of Pandaria) you receive a monetary reward instead of experience. The same rules apply, you get more money if the quest was more difficult.
Effective use of the auction house can be a very lucrative source of income even for low level characters. Many experienced and wealthy players wish to experiment with crafting professions on lower level characters, but they don't want to go through the trouble of gathering the materials in newbie zones. Therefore, you might be surprised by how much you can charge for non-gray items that drop off of low level mobs. Boar meat and linen cloth can net you a lot more money than normal questing. In addition, consider picking up two out of the three gathering professions (skinning, mining, and herbalism) in order to sell the material.
Learning a crafting profession is more fun for many players than gathering, but making money with crafts can be more difficult. Raw materials (herbs, gems, ore, and skins) will often sell for more money than finished products that you can create with a tradeskill (armor, weapons, glyphs, potions, and other items). You can search the auction house to discover how much is a reasonable rate to charge for your items. Remember that prices can change as a response to supply and demand, so it is possible to improve your profits by holding onto your products and timing the market. When you get more experienced in the game, don't forget to investigate useful AddOns to give you that extra edge.
Historically, service professions, like Enchanting were difficult to make money with, but with the addition of vellums to apply your enchants and new tools in the profession windows for selling in the trade channel, these professions have found some new life as money makers.
In addition, there are always the Trade Channel (only in cities) and the auction house (which appear in all racial capital cities, including Orgrimmar, Thunder Bluff, Silvermoon, and Undercity for Horde and Darnassus, Ironforge, Exodar, and Stormwind for Alliance) where you can hawk your wares. All auction houses associated with a particular faction are linked, and there are also neutral auction houses which are also linked. The neutral auction houses are more expensive to post for sale (on the order of three times), and are not linked to either of the primary factions.