My answer is specific to the US because you mentioned the Federal Reserve, but a similar system is in place in most countries.
Do interest rates increase based on what the market is doing, or do they solely increase based on what the Federal Reserve sets them at?
There are actually two rates in question here; the Wikipedia article on the federal funds rate has a nice description that I'll summarize here. The interest rate that's usually referred to is the federal funds rate, and it's the rate at which banks can lend money to each other through the Federal Reserve.
The nominal federal funds rate - this is a target set by the Board of Governors of the Federal Reserve at each meeting of the Federal Open Market Committee (FOMC). When you hear in the media that the Fed is changing interest rates, this is almost always what they're referring to.
The actual federal funds rate - through the trading desk of the New York Federal Reserve, the FOMC conducts open market operations to enforce the federal funds rate, thus leading to the actual rate, which is the rate determined by market forces as a result of the Fed's operations. Open market operations involve buying and selling short-term securities in order to influence the rate.
As an example, the current nominal federal funds rate is 0% (in economic parlance, this is known as the Zero Lower Bound (ZLB)), while the actual rate is approximately 25 basis points, or 0.25%.
Why is it assumed that interest rates are going to increase when the Federal Reserve ends QE3? I don't understand why interest rates are going to increase.
In the United States, quantitative easing is actually a little different from the usual open market operations the Fed conducts. Open market operations usually involve the buying and selling of short-term Treasury securities; in QE, however (especially the latest and ongoing round, QE3), the Fed has been purchasing longer-term Treasury securities and mortgage-backed securities (MBS). By purchasing MBS, the Fed is trying to reduce the overall risk of the commercial housing debt market. Furthermore, the demand created by these purchases drives up prices on the debt, which drives down interest rates in the commercial housing market.
To clarify: the debt market I'm referring to is the market for mortgage-backed securities and other debt derivatives (CDO's, for instance). I'll use MBS as an example. The actual mortgages are sold to companies that securitize them by pooling them and issuing securities based on the value of the pool. This process may happen numerous times, since derivatives can be created based on the value of the MBS themselves, which in turn are based on housing debt. In other words, MBS aren't exactly the same thing as housing debt, but they're based on housing debt. It's these packaged securities the Fed is purchasing, not the mortgages themselves.
Once the Fed draws down QE3, however, this demand will probably decrease. As the Fed unloads its balance sheet over several years, and demand decreases throughout the market, prices will fall and interest rates in the commercial housing market will fall. Ideally, the Fed will wait until the economy is healthy enough to absorb the unloading of these securities.
Just to be clear, the interest rates that QE3 are targeting are different from the interest rates you usually hear about. It's possible for the Fed to unwind QE3, while still keeping the "interest rate", i.e. the federal funds rate, near zero. although this is considered unlikely.
Also, the Fed can target long-term vs. short-term interest rates as well, which is once again slightly different from what I talked about above. This was the goal of the Operation Twist program in 2011 (and in the 1960's). Kirill Fuchs gave a great description of the program in this answer, but basically, the Fed purchased long-term securities and sold short-term securities, with the goal of twisting the yield curve to lower long-term interest rates relative to short-term rates. The goal is to encourage people and businesses to take on long-term debt, e.g. mortgages, capital investments, etc.
My main question that I'm trying to understand is why interest rates are what they are. Is it more of an arbitrary number set by central banks or is it due to market activity?
Hopefully I addressed much of this above, but I'll give a quick summary. There are many "interest rates" in numerous different financial markets. The rate most commonly talked about is the nominal federal funds rate that I mentioned above; although it's a target set by the Board of Governors, it's not arbitrary. There's a reason the Federal Reserve hires hundreds of research economists. No central bank arbitrarily sets the interest rate; it's determined as part of an effort to reach certain economic benchmarks for the foreseeable future, whatever those may be.
In the US, current Fed policy maintains that the federal funds rate should be approximately zero until the economy surpasses the unemployment and inflation benchmarks set forth by the Evans Rule (named after Charles Evans, the president of the Federal Reserve Bank of Chicago, who pushed for the rule).
The effective federal funds rate, as well as other rates the Fed has targeted like interest rates on commercial housing debt, long-term rates on Treasury securities, etc. are market driven. The Fed may enter the market, but the same forces of supply and demand are still at work. Although the Fed's actions are controversial, the effects of their actions are still bound by market forces, so the policies and their effects are anything but arbitrary.