Throughout history, humankind has traded. At first, we bartered, but in time we created a more universal tool to make trading easier: money.
The Rise of Currency 
Currency is defined by Webster as a medium of exchange. Almost anything can be used as currency provided that people are willing to accept it in exchange for goods.
There are a few simple reasons why money, currency, would be used to trade with. First of all, its value is agreed upon and can be used to measure the price of other items. That is difficult to do with livestock or other things. The second reason is that it survives and looking at the example of livestock trading, doesn't die, which means that it can be saved. Finally, a medium of exchange can be carried around easily.
The Advantages of Currency 
To explain the advantages of currency, we'll take an example of trading with cows and trading with coins. The following is a short story illustrating the differences.
One day, a man in the market offered to trade his two pigs for one cow. Because I had a cow and needed two pigs, I approached the market-man to make a trade. The man looked at my cow and rejected it. He told me it was too old. I told him I would bring my other cow tomorrow. I trekked home.
The next day, I went back with my younger fuller cow, but unfortunately, when I arrived, the cow had dehydrated. I was now without a cow, and the market-man wouldn't give me anything. I went back home.
The next-next day, I went to a different market with 20 coins and my old cow. Someone offered to trade two pigs for 15 coins. I gladly accepted and he took a look at my coins to ensure their authenticity and accepted them. I had 5 coins left and I decided to save those. I sold my old cow for 8 coins even though I think I should have gotten at least 12 coins. I hurried home with my two pigs and 13 coins.
...In the above example, we can see how easy it is to use currency to facilitate trade.
The (Often Hidden) Disadvantages of Currency 
When man relied solely upon the barter system mentioned above, any and all trade had to be considered "fair" to both parties. However, with the introduction of a government-controlled monetary system, the above transactions would be fair to neither.
The seller above had 20 coins from previous business and 8 coins for his cow for a total of 28 coins he had obtained. Having spent 15 coins on pig he went home with two pigs and 13 coins. In a fully regulated system, money acts as the mechanism for tax collection. The seller, who earns in a 50% or so tax bracket owes 14 coins to the government. Alas, the poor soul arrives home already indebted to his government.
The person who received the money, of course, needs to treat it as income and give half of it (7 1/2 coins) to the government for his taxes. Thus in a series of 5 or fewer transactions, all money (thus work) is successfully transferred to the government.
Definition of Currency 
(1) an agreement within a community to use something as a medium of exchange. Some examples: dollar bills, gold coins, frequent flyer miles, Japan's "fureai kippu" (caring relationship tickets), pogs, Ithaca hours, Bali's "narayan banjar" (meaning work for the common good of the community);
(2) a unit of exchange or some thing, which in itself may have no fundamental utility to us, but which we believe can be exchanged in the market for something that does. (i.e., what we use to buy things from someone else as opposed to bartering or taking by force);
(3) a unit of account (i.e., the measuring stick of value);
(4) a store of value (i.e., as a way of holding wealth);
(5) a standard of deferred payment for current goods.
(6) a unit as above that then carries information between the buyer and the seller. That information then "signals" the invisible hand of the marketplace to produce more goods and services of one type at the expense of another type. If I am a farmer who invests 2 gold coins to grow corn that I sell for 20 gold coins, then I am encouraged. If I invest 2 to sell for 1, I am discouraged. If my neighbours and others in the marketplace see me profiting in corn, then they will do the same (i.e., the marketplace delivers more corn). When they sell their corn and compete with mine, the price goes down (i.e., the marketplace devalues our contribution). When I have a choice to grow corn or wheat, I am "greedy" and try to maximize my returns. Thus, the marketplace encourages me to grow wheat with high prices and discourages me from growing corn with low prices. Thus, it sacrifices corn production for the wheat production. The marketplace allows the consumer to control the means of production to deliver the optimum amount of corn. If the value of money is stable over time, the cost of commodities declines over time for a variety of reasons (e.g., I get really good at growing corn; I invent more ways to produce more corn at a lower cost).
Types of Economies 
There are generally three types of economies; free-market, mixed, and planned. Mixed markets are so-called free-markets, capitalism, etc. However, few deny that it is impossible to eliminate government interaction with the economy even in mixed markets. Planned economies are communist (Marxist, Stalinist, etc.), fascist, cooperative, etc. — economies where the government plans, or commands, the direction of the economy.
Free-market Economies 
Free-market economies are where there is private ownership of the factors of production and owners have the right of exchange. The advantages of the free-market economy are that competition helps eliminate the worst products and services, allowing only the best to thrive. This is very similar to the survival of the fittest concept in the theory of evolution/natural selection in biology. Open markets allow imports from other countries to compete directly with the native country's businesses. External competition also promotes survival of the fittest.
Mixed Economies 
Mixed economies are where a market exists but is restricted by the government. In general, the market is restricted to enact social engineering and/or redistribution of wealth. As previously noted, it is very difficult to remove the government from the equation once they are in place.
Planned or Command Economies 
These types of economies are the mainstay of totalitarian governments such as those of Cuba and Cold War Russia. Though China was a command economy, the government there has slowly relaxed this and has allowed the growth of private business. The main feature of command economies is nationalization of businesses.
Political viewpoints are usually aligned with certain economic types. The left-wing or liberal viewpoint generally supports government intervention in the economy to make things fair, and at times also supports the nationalization of certain industries. The right-wing or conservative viewpoint, on the other hand, generally supports privatization, the opening of markets, less government intervention, etc. The libertarian viewpoint advocates a market completely free of governmental restriction, while the socialist would argue for total governmental control of business.
It is important to note that while each ideology has a particular economic view, many eventually turn to mixed economies. Socialist countries, for example, are mixed, as are capitalist/conservative countries. Even communist countries are mixed economies; a good example is that of China, noted previously. Thus, the views of advocates of the extremes (completely planned or completely free) must be taken with a grain of salt.
Models of Economics 
The Supply and Demand model is widely known.