What is the significance of interest rates? What happens when the Federal Reserve fiddles with them? Who is affected and how when the rates are raised?

Interest, in lay terms and broadly speaking, is the price we pay to borrow money.  Theoretical economists recognize that interest is comprised of three inherent parts: 1) an "originary" rate, which reflects people's time preferences.  This refers to the intensity with which we prefer funds in the present over funds in the future.  In other words, to get people to part with money they have today to lend to you, you must pay them at least enough to overcome their preference to spend their money now.  2) debtor's risk premium, which reflects the riskiness of the loan because of the chances the borrower may default.  If people started defaulting on their loans in large numbers for any reason, it would cause interest rates to rise because lenders would not lend without enough "extra" to cover the increased risk of default.  3) inflationary risk premium, which reflects the general expectations about price inflation.  The higher the inflation, the higher interest rates will be.  No one will want to lend money at 10 percent if inflation is soaring at 50 percent.

At any given time in a free economy, these three psychological factors come together to create both the supply and the demand for lendable funds, or money.  The interest rate is the "price" which brings supply and demand together.  It changes as people save more and hence put more funds on the market to lend.  More money on the market, in and of itself, would tend to cause interest rates to fall.  But if people saved less and therefore made fewer funds available for lending, that would tend to cause interest rates to rise.  On the demand side, if people see great opportunities for investment because the economy appears sound and growing, they would tend to demand more funds, which would tend to cause rates to rise (but of course, as those rates rise, they also act as an incentive to savers to save more, which keeps the rate from rising even further).

When markets are free, changes in interest rates occur rather quietly and beneficially, reflecting true market conditions of the supply and demand of money.  However, in today's mixed economy—where government controls the money supply—central banks are actively manipulating interest rates all the time.  When the central bank increases interest rates, it does so by withdrawing reserves from the banks.  It does this usually when it intends for the money supply to fall, or at least rise at a slower pace.  When the central bank cuts interest rates, it does this by injecting reserves into the banking system and it does this usually when it intends for the money supply to increase.

What would be the effects of an increase in interest rates?  It all depends on why those rates are going up.  If it is because there is strong demand for funds because the economy is strong, then the higher rates would tend to increase savings.  If rates are going up because prices (price inflation) is soaring, then rising rates will generally put a crimp on the economy and sow the seeds for a recession as the painful "price" we will have to pay for engaging in the previous inflation.

That's a "macro" view of your questions.  For a "micro" view, let me say that a hike in interest rates generally tend to hurt those sectors of the economy that are especially interest-rate sensitive because they depend heavily on borrowed funds—housing and construction and auto production, for example.

"When markets are free, changes in interest rates occur rather quietly and beneficially, reflecting true market conditions of the supply and demand of money."

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