Nope.  If you just put a zinc strip and a copper strip into a beaker of water, nothing happens, correct?  If you add some sodium bicarbonate there, nothing happens, right?  But now if you add some copper sulfate into the solution, you suddenly see a layer of copper forming on the zinc metal.  Why is this?  This is because the copper ion is a good oxidizer in comparison to the zinc metal.  It will oxidize the zinc (Take electrons from it) and form copper metal while the now positively charged zinc ions move into solution.

In chemistry, EVERYTHING involves the movement of electrons.  When the copper ions turn into copper metal, the following is happening:  Cu(+2) + 2e- => Cu(s).  At the same time, the zinc metal is giving up electrons to form Zn(+2) ions in the following reaction:  Zn(s) => 2e- + Zn(+2).  Both reactions (Half-reaction) occur at the same time, so the overall reaction you see is Zn(s) + Cu(+2) => Cu(s) + Zn(+2).  If you take a beaker with a ZnSO4 solution in it and place a zinc rod in there, and in a separate beaker place a copper rod in a solution of CuSO4 you can actually carry out each of the half reactions.  In order for the reaction to get going, you need to go and give the electrons a way to get from Beaker 1 into Beaker 2, and a way for the charges to balance out.  To get the electrons to flow, you connect the two rods with a wire.  To get the charges to remain balances, you add a "salt bridge".  The salt bridge is just a tube with a salty solution in there that is plugged on the ends with cotton, or a semi-permeable membrane to allow the ions to flow through, but not the solution.  When you do this, the half reactions I wrote above happen spontaneously!  

Because the copper ions are a relatively good oxidizer, it tries to pull electrons from the copper metal rod.  The thing is, the copper metal doesn't want to give up the electrons.  Since it is connected to the zinc rod, however, it is able to pull some electrons from the zinc and give them to the copper ions in solution.  As a result, electrons flow from the beaker with the zinc rod into the beaker with the copper rod.  Cu(+2) ions grab the two electrons and form copper metal while the Zn rod gives up two electrons and forms Zn(+2) ions.  Because there is a movement of electrons you can measure a flow of current.

So why doesn't Na and Cl2 form when Zn and Cu strips are placed into a solution of NaCl?  In order for the cell to be spontaneous, there has to be a source of zinc metal, zinc ions, copper metal, and copper ions present.  This is because the Cu(+2) ions are good enough oxidizers to pull the electrons away from the zinc.  The Na+ ions in solution don't have any electrons to give up, and the Cu(+2) ions are NOT a strong enough oxidizing agent to pull an electron off of the chloride ion.  As a result, there is no way to transfer electrons so no current flows and no reaction happens.  If you put some ZnSO4 in with the Zn, and CuSO4 in with the Cu, then the favorable reaction will be for the Cu(+2) to take electrons and for the Zn(s) to give up electrons.  The sodium ions are definitely not a strong enough oxidizer to take electrons over the Cu(+2) ions, nor does it have any electrons to give up on the Zinc side of things.  As a result, Na+ just sits around doing nothing.  At the other end of the spectrum, Zinc metal is much more easily reduced than Chloride (Cl-) is, so when the copper ions call for some electrons from the other beaker, it will take it from the Zinc metal and not the chloride ions.  As a result, Cl- stays in solution as a spectator.  The Copper ions are simply not a strong enough oxidizer to take any electrons from chloride, and sodium ions are not strong enough reducing agents to give up any electrons to copper.  

In addition, even if a power supply is used, you won't be able to make ANY sodium metal from a solution of NaCl because Na+ is a far weaker oxidizing agent than H2O is and the water will take the electrons and subsequently decompose instead of the Na+ taking an electron and forming Na(s).  If you use a mercury cathode then you can do it as the sodium forms a compound/amalgam with the mercury, but it's a pain in the ass to then try and separate the sodium out without losing it all to oxidation.

Nice explanation, but you can leave out the ZnSO4. The CuSO4 is essential, because you need Cu(2+) ions. The ZnSO4 is not essential. In fact, any inert ionic compound, which does not react with the zinc electrode works. The zinc electrode still gives off its electrons to the copper (through the wire) and Zn(2+) goes into solution.

So, you can take one beaker with any inert soluble salt (e.g. Na2SO4, NaCl), a salt bride between the beakers and the other beaker containing a copper salt (CuSO4 is perfectly suitable).

Thank you.    Yeah, it's true that any inert soluble salt will work, but it's just good practice to use a salt of the metal that composes the electrode.  This way you remove ALL variables in regards to the other cations in solution there.