Lets look at this a little:
There are two types of cells:
First lets discuss the galvanic cell
The galvanic cell, also called a voltaic cell, (a battery being discharged by a load is a galvanic cell).
(Note: the battery being charged would be an electrolytic cell, a totally different type of cell we will discuss below).
This galvanic or voltaic cell produces a current, thus a voltage, energy is given off from an electrochemical reaction, electron transfer between two dissimilar metals in solution, which generates current through an external wire, electrons move electrochemically through the cell and electrons travel through the external wire which can light a load like a light bulb, this external current being generated by the chemical reaction of two dissimilar metals in an electrolyte.
When we cement silver from a silver nitrate solution, using a bar of copper metal we are making a galvanic cell, although the electrical energy produced by the electron flow here is just wasted as heat into the solution, the copper being Higher in the reactivity series of metals has all of its electrons, and will donate electrons to the to the dissolved silver Cations in solution, the copper is oxidized (lose of electrons), and dissolved into solution and the silver is reduced (gain of electrons) the solution becomes copper nitrate as the silver cements out of solution, so this can also be considered a chemical displacement reaction.
Another type of galvanic cell, this cell is two dissimilar metals in an electrolyte solution (the electrolyte can be acidic or alkaline), these dissimilar metals are connected either connected by an external wire, or wires connected to a load like a small light bulb or voltmeter. In this cell with two dissimilar metals, one becomes the anode (the more reactive metal). On the other side of the cell the metal becomes the cathode, there is an electrochemical reaction going on inside the cell, electrons begin to be transferred, there is a potential difference between the different metals in the electrolytic solution, that will cause a reaction and electron transfer. The anode will be oxidized (loss of electrons) (this lose of electrons makes it the anode of the cell or our battery (-) or the electron poor electrode, {the wire from this anode is moving electrons through the external wire, through the lamp load to light it, and to the cathode to make the cathode electrode electron rich (+)}, this anode electrode metal will loose electrons and go into solution with the electrolyte. the metal Cation joining with the surrounding electrolyte anion to form a dissolved metal in solution.
The other electrode on the other side of this galvanic cell (+) the cathode now with excess electrons (from the wire through the load, becomes electron rich or has excess of electrons, so it will easily donate electrons to the metal Cation in the surrounding electrolyte, reducing the electrolyte Cation (gain of electrons) back to metal, so this metal Cation that has now gained back an electron to change back to metal is able to plate out of solution onto this cathode,
So there is an electron movement inside the cell or battery moving electrons electrochemically, generating an external electron flow through the wire and lighting our light bulb or moving a needle on our voltmeter.
Note: this galvanic cell is a battery discharging producing a current through a wire or through a load.
Note: (a battery being charged is not called a galvanic cell, the current is forced backwards through the cell or the reaction is forced by an external source {battery charger, power-supply, or a bigger battery}{the battery being charged is a different type of a cell we will discuss below called the electrolytic cell).
Now lets discuss the other type of cell:
The electrolytic cell:
The electrolytic cell has many uses purifying metals, electro-winning metals for recovery, a battery being charged, and electroplating, or making chemicals through electrolysis…
This is the type of cell we use to purify our silver, this is also the type of cell (a battery being charged), or used to electro-winn copper to make impure copper into pure copper.
The electrolytic cell does not make current or voltage like a battery or the galvanic cell, it does not produce electricity, or energy (like the galvanic cell above does), the electrolytic cell uses or consumes electricity or energy to force an electrochemical reaction that may not normally occur spontaneously, within the cell from an external power source (battery charger or DC power supply, or from and external battery or a battery of higher voltage than the battery being charged, or higher than the potential difference of the reaction we are forcing). The electrolytic cell consumes energy.
Here the positive of the power supply is connected to one electrode, which is now called the anode (+).
The negative of the power supply is connected to the other electrode now called the cathode (-).
Oxidation occurs at the anode (lose of electrons), and reduction occurs at the cathode (gain of electrons), the metal from the anode is oxidized losing an electron is dissolved into solution with the electrolyte anion, at the other side of the cell the electrolyte surrounding the cathode (where the cathode has excess electrons given to it from the external power supply), the Cation metal of the electrolyte is reduced to metal (gain of electrons) this metal Cation now gaining an electron and changes back to metal which plates out of solution onto the cathode freeing the anions of the electrolyte to move back to the anode compartment to pick up another oxidized metal from the anode.
So at the anode (electron poor electrode), the metal is oxidized (lose of electron) and dissolved into solution as an ion (a Cation), which joins with an Anion and becomes the electrolyte solution.
Note (+) and (-) polarity's of the cells in the descriptions above, note also in a battery being discharged (galvanic cell), and the battery being charged (electrolytic cell) this polarity changes, so basically in the battery the same electrode can have either (+) Or (-) polarity depending on if it is being charged or discharged, this gets a bit confusing for me to explain, in the galvanic cell the anode is (-), but in the electrolytic cell the anode is (+), but in both of these cells oxidation (lose of electrons) always occurs at the anode, and reduction (gain of electron) always occurs at the cathode.
The galvanic cell produces energy.
The electrolytic cell consumes energy.
Note: solar_plasma, when you connected the two dissimilar metals together by a wire (you were calling them the cathode) in your electrolytic cell.
You just made two different types of cells in one pot, with two different reactions going on simultaneously in this same pot, two different electrochemical reactions, and independent of each other.
The main normal electrolytic cell, and the galvanic cell made up of the cooper and silver tied together by a wire making a battery, or galvanic cell (the two electrode you called cathodes)
These two cells moving electrons independently, in the same pot, two current movements, one being forced into the cell (the electrolytic cell electrons forced to move by the power supply) and (and a galvanic cell the spontaneous reaction of two dissimilar metals connected by a wire in an electrolyte), so you also made a galvanic cell inside the electrolyte, this galvanic cell or battery is making a reaction and a voltage inside your electrolytic cell that is independent of the other reaction of your electrolytic cell. you made an independent battery inside your cell by the two dissimilar metal connected together by your (common cathode wire) so the pure copper and silver and (" cathodes") are actually becoming an galvanic cell themselves moving electrons between themselves, independent to the other reaction in your "electrolytic cell" (the impure silver solder anode and the two "cathodes" which is another reaction and another type cell all together, and having its own reaction going on simultaneously.
So in the galvanic cell, the copper would become the anode, and would be oxidized (with copper going into solution, the silver electrode would essentially become the cathode (or common cathode of both of your cells in this pot.
this was a bit hard to explain.
Edit my mistake (Thanks frugalrefiner, for catching it )