I typed this in response to a seemingly simple question concerning electrochemistry, but it turned into a substantial walk through the origins of chemistry to practical examples. Since it's so long, I decided to save it here on my website.
Fundamentally, chemical reactions -- we're talking ionic here -- are driven by electron affinity. It always takes energy to remove electrons from an atom (to ionize it) -- that should be obvious, because a neutral atom is preferred to a negative electron some distance from a positive core. Charge seperation takes energy. Less obvious is how adding an electron to an atom is often preferred. This is especially true of the elements on the right hand side of the periodic table: oxygen, fluorine and chlorine are the most important. The reason is purely quantum mechanical: electrons fit into these atoms' vacant orbitals, and atoms like filled orbitals. Remember, adding a negative charge to a neutral body makes absolutely no sense from a classical standpoint!
Since ionization always costs energy, how is it that we have any positively charged ions? Well, some atoms have more electron affinity than others, and some have less ionization energy than others. These electronegative elements are able to pull the electrons off the electropositive elements, forming ionic compounds.
Now to segue into electrochemistry.
Electrochemical cells are typically made with water, which contains oxygen, an excellent electronegative element. Sure, water is electrically neutral, and already bonded -- two hydrogens are sharing their electrons with the oxygen, making a pretty stable, neutral, covalent compound. But it's not perfect, and about every 10 millionth water molecule splits up due to thermal energy, forming H+ and OH- ions. Let me put it this way: chemicals are like people in bad relationships -- most of them stay together, but a lot go around continually divorcing partners. The physical world is just as seamy as society at large!
Those H+ ions would love to get some electrons, and in neutral water, they recombine with OH-'s all the time. Now, if you dropped a chunk of zinc in the water, some of the zinc dissolves, putting Zn2+ ions into solution. This is written as a half-cell reaction:
The electrons go into the metal, so the solution becomes positive and the metal negative. Charge imbalance is a really bad thing, so it's a self limiting reaction. The equilibrium depends on how much the solution wants to dissolve, which since the solution is always the same hydrogen and oxygen stuff (we're talking pure 20°C water at 105 Pa pressure), standard values depend on the ionization energy of the metal added. (And yes, metals have different potentials against different solvents.) It turns out water is able to generate about 0.8V against zinc, 0.5V across iron, and none against copper (copper is a noble metal). More about that one later.
If we remove charge from the solution, we can neutralize the standoff between zinc and water and make some real current flow. Say we put a strip of platinum into the solution: nice and inert, this metal will not react with the cell. Say we short the zinc and platinum together: now the platinum has the same potential as the zinc, which is about -0.8V relative to the solution. Now what? The platinum attracts positive ions to it, like H+. It would be nice if the platinum would give up a few electrons, so the system can become electrically neutral again. Ah, but can it? Let's see...what if we did...
Heyy, that seems to work, making hydrogen gas. And it turns out this reaction is defined as 0.000V in water, so against the -0.8V we've got, it will most certainly proceed. Whaddya know, we've got a battery!
Back at the zinc surface, OH- ions are moping around. To keep the solution electrically neutral, it would be wonderful if Zn2+ neutralized with 2 OH-'s, and this certainly happens. The product doesn't have to be insoluble, but Zn(OH)2 does happen to be quite insoluble. This reaction is a surface thing, and it turns out zinc doesn't corrode nearly as quickly as you might expect because it coats itself with this gunk -- it would love to react, but there's just no circuit to proceed. Aluminum (Eo ≈ -1.7V), magnesium (Eo ≈ -2.3V), titanium (Eo ≈ -1.6V to Ti2+) and many others would burn violently if not for this same protective layer. (When the layer isn't given a chance to protect, these metals are as reactive as you would expect: flash powder contains powdered aluminum, and burns so fast, large amounts can go high-order and detonate!) This frustrates battery manufacturers, so they have devised ways of getting around it: a mercury amalgam on the reaction surface is one reliable method, keeping fresh metal always exposed without sacrificing shelf life.
Practical batteries usually have fuel and oxidizer mixed into one self-contained unit. Releasing hydrogen gas isn't very practical, but burning it is. The ancient "dry cell" is made of a paste of conductive electrolyte (usually ammonium chloride, which does not participate in the reactions, it just carries charge), graphite (also conductive and inert, but solid) and manganese dioxide, which is a powerful yet stable oxidizer. Nascent hydrogen generated at the cathode (usually an inert graphite rod) is oxidized by the MnO2, producing Mn2O3 (a reduced form) and an extra fraction of a volt. The two half-cell potentials add to give the usual terminal voltage of 1.5V seen on these types. Because MnO2 is so insoluble, it does not spontaneously react with the zinc anode, giving these cells a useful shelf life.
An older sort of oxidation is had by putting an easily-reduced ion into solution. Examples include ferric chloride, copper sulfate and others. Earlier, I mentioned copper is a noble metal: it does not dissolve in water very much, because water doesn't have enough affinity for its electrons. If you supply a reverse bias, you can force its electrons out anyway, putting copper ions into solution. (Copper typically forms the doubly-ionized Cu2+ as zinc does, but the singly ionized Cu+ is also well known.) Now, you may ask, why would copper want to dissolve in water? Can't water just reduce it? Remembering this reaction:
Ah, but that requires hydrogen gas, and some way for it to react (H2 doesn't dissolve in H2O very well, and reacts with it even less. A platinum mesh is usually used to catalyze this reaction. Lovely platinum shines again!). So we can't use this as an excuse for copper spontaneously dropping out of solution. That's why it doesn't!
In fact, water is quite stable against oxidation. An example is:
Fluorine gas is an insanely strong oxidizer (it REALLY wants electrons, to the tune of 2.89 volts!), so it rips the hydrogen off the oxygen, leaving it to mope around and form oxygen gas. It takes a potential of 1.229V to oxidize water, whereas copper has only +0.34V, which is a rather mild oxidizer.
Substances are said to be oxidizers if they can oxidize another substance (or be reduced by it), while reducing agents are capable of reducing another substance from solution (or that substance oxidizes it into solution). Hydrogen is the arbitrary absolute standard used in solution making for a convienient zero basis, but redox reactions are always relative; copper, a mild oxidizer itself once ionized, will displace silver from solution, because silver ions are a better oxidizer than copper ions; or equivalently, copper metal is a better reducing agent than silver metal.
Say we have a solution of copper sulfate in water. The cathode can be anything inert: copper would certainly make sense. Our old standby, platinum, would be on the expensive side, but would certainly suffice. Let's say we drop in a zinc anode: immediately, some zinc dissolves, giving the metal a negative charge. Positive copper ions are attracted to it and suck electrons from the zinc, depositing copper metal on the spot. This is called cementation, long used to deposit less reactive (often more precious) metals from solution: the platinum group metals, gold, silver, etc.
Now we have a copper plated zinc anode. Huh, that's kind of useless. Hey, what if the copper doesn't completely coat it? What if it flakes off? Rats, we've got a not-battery that's discharging itself! ...Right? Ah, but lucky for us, copper sulfate solution is dense, and if the zinc is held high in the electrolyte, it will only react with the copper at that height. As long as it isn't stirred, diffusion limits the reaction. Now that things have calmed down, we can connect up a circuit. But wait, if there's no copper at the anode, how is it going to work? Reduction occurs at the cathode, remember? The nice thing about a cell is, the actual "burning" of the anode is seperated into two steps, reduction and oxidation, and because they take place on seperate electrodes, we can make use of the current between them.
Now our cell, called a gravity cell (for the density seperation), is nice and orderly and we can draw current from it. Supposedly, these sorts of cells were useful in backup duties, such as for telecom supplies, where a low average current (tens of mA) was needed. Obviously, they can't be disturbed very much, which isn't a problem in static use, sitting on a basement shelf somewhere. Maintainance is very easy: scrape out the copper sludge in the bottom, replace the nub of an anode with a new ingot of zinc, and toss in some more copper sulfate crystals. (Of course, zinc sulfate needs to be removed too, eventually.)
So there you have it, some chemical and electrochemical fundamentals. It would be nice to say that much of this can be derived from first principles, which is how I've written it, but the funny thing about chemistry is, there are always twists. Chemistry works on well defined physical principles, but sometimes there are so many principles competing, it's hard to predict which ones are the most important. Guidance is perhaps best left to empirical evidence.