In an electrochemical cell, we are running a redox reaction, but we have physically separated the oxidation and reduction reactions into different locations. These take place in different "halves" of the cell.
We then connect the two halves of the cell with an external wire to allow for electron flow from one side to the other. We also connect the two halves with a "salt bridge" to allow spectator ions to flow to counter balance the electron flow.
The oxidation and reduction reactions are now in different locations. To allow the electrons to flow from the oxidation reaction (that is producing electrons) to the reduction reaction (that is consuming the electrons), we need to connect the two reactions electrically. This is accomplished by using metal electrodes. These electrodes serve as the place where the chemistry is taking place. Sometimes they are part of the reaction. Sometimes they are simply used as a means to "deliver/collect" the electrons. The electrode in the oxidation reaction is called the anode. The electrode for the reduction reaction is called the cathode. The electrons flow from the anode to the cathode.
Let's look at a picture of the cell below which contains zinc metal and zinc ions on the oxidation side and copper metal and copper ions on the reduction side.
The left hand side is a picture of the anode half of the cell where the oxidation is taking place. The solid piece of zinc is the anode. The righthand side is the reduction half where the copper 2+ ions are being reduced to copper metal. The solid piece of copper is the cathode. The two sides are connected by a wire that allows electrons to flow from the anode to the cathode. They are also connected by a salt bridge. Since negative charge is flowing from the anode side to the cathode side, some spectator ions need to compensate this flow (or the cathode side will end up with lots of extra negative charge). So in the salt bridge either the positive sodium ions could flow from the anode side to the cathode side or the negative nitrate ions could flow from the cathode side to the anode side. Or both could happen.
For some reactions, neither the reactants nor the products are a metal. In this case, an inert (non-reactive) metal electrode is required to deliver/collect the electrons for the reaction.
For example, in the cell below at the anode, Fe2+ is oxidized to Fe3+. This reaction occurs at the surface of a Pt electrode. Also, note the ions in the salt bridge could be any strong electrolyte. They don't have to participate in the chemistry.
\[\rm{2Fe^{2+}(aq) \; + \; Cu^{2+}(aq) \rightarrow 2Fe^{3+}(aq) \; + Cu(s)\; }\]
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