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Well first thing you need to understand is that "money" is a very broad word that includes lots of different things that enter the system in different ways.

Cash enters our system through the banks, you put money in the bank in some form (maybe cash, maybe an EFT from your direct deposit, maybe some other way). Then later you go to the bank to get some of your money out as cash, meanwhile behind the scenes the bank has sent old beat up cash back to the Fed for destruction and the Fed has sent them shiny new cash, the cash you get might include some of that shiny new cash, or it might be some not so shiny but still usable cash. Either way the cash gets into circulation primarily by replacing money that was already yours before it was handed to you.

In the more general stroke money enters circulation through wealth expansion. We produce wealth every day in this country by harvesting raw goods (wood, stone, metal, food) from the earth. Raw goods are turned into refined goods which causes them to gain in value thus adding more wealth. At various stages in this process these things are sold for some form of money, maybe cash but probably a check or EFT.

There are also Federal Reserve loans which bring money, but not cash, into the economy. Though not all loans stem from the Fed as a source.

The important thing to remember is that the vast majority of money that circulates through our economy never has any level of representation that is produced by the Fed. It moves in checks, in EFTs, in stock certificates, in travelers checks (which are different than regular checks), in debit cards (which is kind of an EFT but not really), or various other forms I don't remember.

Here is a simple answer from one of my favorite websites:
http://en.wikipedia.org/wiki/Debt_levels_and_flows

Debt is used to finance and pay for undertakings and business around the world. Debt levels are worth 3 years of GDP in many countries that has an annual GDP/person above $10,000. Global debt levels are perhaps worth two or three years of GDP. GDP (at currency exchange rate) was $40 trillion during 2004. Debt levels may therefore be about $100 trillion.

$5.7 trillion of debt was issued in 2004 according to Thomson Financial numbers, while GDP grew $4 trillion (currency exchange rate). That does not mean that debt grew faster than GDP on a global average (even if it has done so for years after 2001 in the USA). Debt is often issued with a repayment plan (a "time to maturity" in some cases), repayment times may be between a few days (interbank cash flow management) and 50 years or longer (consumer real estate debt). The average repayment time of all global outstanding debt is perhaps 10 years.

When debt matures new debt is many times issued to repay the old debt, perhaps from the same creditor. That is one reason why debt issuance far surpasses equity issuance in currency value. Equity is another way of financing business, it has no set time to maturity, and pay no set interest, it pays profit from the company it is a claim on.