It's said that "money makes the world go round." This isn't literally true, but money can be pretty important, even in the world...of warcraft.
The basic unit of currency in Azeroth is the gold piece. It takes 100 silver to make 1 gold. It takes 100 copper to make 1 silver. Early on, your character will pick up a few pieces of copper here and there. Later that will become silver, and eventually gold. As your money-making capabilities increase, so will the costs of the things you want.
It is important to note that all money will stay on your character, and that you cannot lose money. Being killed does not cost you money directly (or make your money lootable by other players.) There is no need to store money in the bank, so it isn't possible. Also, any time you pick up more than 100 copper, it will automatically convert to silver, and the same is true for silver into gold. There is no unit of currency larger than gold.
Money is important because of the many different places it can be spent. This includes:
Money can be earned by many difference ways as well. This includes:
The best way to earn money is to pick up the gathering professions - mining, skinning, or herb gathering. By selling the materials you gather on the Auction House you can earn a decent amount of gold to support you financially for your mount, and even epic mount. Also questing is a very good way to accumulate a decent amount of gold while getting good xp for leveling. And keep in mind, you should spend most of your time leveling and not worry much about earning gold, because once you are 70, you will be able to earn gold at a much faster rate.
For earning gold at level 70, read the article Gold Farming.
The Auction House and the Trade Channel provide you with a valuable means of selling your goods and services to other players who need them. If you collect a large stack of cloth you don't need, it's better to try to sell it to another player than to the merchant!
With the exception of exceptionally rare items and services like Enchanting that can't be sold on it, it is usually better to try the Auction House for your sales. Many more players will check the AH in 24 hours than read your message in the chat channels, and there would be way too much spam if that was what everyone did.
Of course, the auction house is a great way to lose money as well, but there is also the potential to "play the market" by buying low and selling high. If you learn the economy of the server, you might be able to turn a nice profit.
Real-Money Trade, or RMT, is the process of paying someone real world money for in-game money. This money is normally supplied by farmers and bots that tirelessly kill mobs to sell the drops. Putting ethics aside, many players think it more manageable to pay for the equivalent of a few hours of their salary than to play the game for many more hours.
However, RMT has a profound effect on server economy, and can also be disruptive to the players in a particular area. Many players also feel that this process is cheating. Purchase of game money or any other in-game services with real money is a bannable offense.
Another "convenience" some players will pay for is power leveling. This involves contracting an outside party to play your character for you to achieve a higher experience level. This is often paired with purchasing gold.
Players quitting the game will also sometimes try to sell their account to someone who quickly wants a class that the quitter played. It is technically against the Terms of Service to even share an account with a family member (except a parent-child relationship,) and although Blizzard does not pursue the transfer of character accounts as diligently, this is still a bannable offense if you are caught. WHICH IS TOTALLY ILLEGAL!