Batteries Do Not Supply Electrons? I've read that a battery does not supply electrons, it establishes the electric field that exerts force on the electrons in the wires.
This makes total sense to me for AC, but not for DC. The way I picture DC is a force is applied and the electrons in the wires move in one direction, but if no electrons are supplied how could this go on once the electrons in the wires are gone? Thank you.
 A: A battery is both a sink and a source of electrons. 
It provides no net contribution of electrons to the external circuit, however.

In the below schematic of an alkaline battery, which is a representative battery configuration:
#3 is the metallic zinc anode
#4 is a separator that conducts ions, but not electrons
#5 is the nonmetallic manganese oxide cathode
#1 and #9 are the positive and negative terminal connections, respectively
#2 and #6-#8 are structural elements not relevant to the question

(Public domain image; click to enlarge)
Chemically, if you took powders of metallic zinc and manganese oxide and mixed them together, the manganese oxide would pull electrons away from the zinc, forming positively-charged zinc ions and a different ("reduced") form of manganese ions.  There would also be some rearrangement of the counter-ions in the mixture, with some negatively-charged anions (primarily hydroxide ions) shifting to be more closely associated with the now-positively-charged zinc ions.
The same reaction occurs in an alkaline battery, except that the ion-conducting separator (#4) only allows the 'ion rearrangement' part of the reaction to occur inside the battery housing.  (Hydroxide ions, for example, diffuse readily through the separator.)  The separator blocks the direct transfer of electrons from the zinc metal to the manganese oxide, however, and so the only route they have available is by moving out through the negative pole (#9), through the circuit, and back in through the positive pole (#1).
Thus, the zinc anode is a source of electrons, and the manganese oxide cathode is a sink for electrons.  Therefore, while there is no net production of electrons from a battery, some of the electrons passing through the circuit almost certainly originated from within the battery.
A: Think of a battery as a parallel plate capacitor. If you put an electron or ion between the plates of the capacitor. The charged particle will move from one terminal to another.
Similarly, in a circuit, the electrons or ions that are put in between these plates are called charge carriers. These charge carriers are found within the conductor or semiconductor of the circuit. For instance, there are free electrons which moves through the lattice in a conductor. When a battery is wired up to this conductor, these free electrons will get "pushed" and they will move inside the conductor with a velocity called drift velocity. However, not every conductor has the same number of free electrons per unit volume. That's where the intrinsic conductivity of a conductor plays its role. The density of free electrons in a conductor determines that conductor's electrical conductivity. Therefore, if there are more free electrons in unit volume inside a conductor, then that conductor is said to have low resistance.
Therefore, as you can see supplying free electrons is an intrinsic property of a conductor. What a battery mainly does is to apply an electric field to these free electrons and make them transmit the electrical energy through the circuit.
P.S: Some charged particles of the battery will end up flowing through the circuit that is shown by Brian's excellent answer. However, the electrical energy provided by the battery is not due to these particles but the potential difference. This flow is a consequence of that.
A: An object which gives off more electrons than it takes in will become more positively charged relative to everything else in the universe than it would have been otherwise, making it harder for anything to draw electrons off of it.  Likewise a device which takes in more electrons than it gives off would become more negatively charged relative to everything else in the universe, making it harder for anything else to push electrons onto it.  The total imbalance can't get very big before the it becomes almost impossible to increase it further.
The reason a capacitor works is that the presence of a positive charge
near one side of a plate will cause all the electrons to flock to that
side, thus making it easier for the rest of the plate to accept electrons
from elsewhere, while the presence of a negative charge near a plate will
tend to push more electrons into the rest of the plate, making it easier
for them to get drawn off.
A battery, unlike a capacitor, has electrons move from one electrode to another.  Typically, the chemistry of the battery is such that either electrode can support chemical reactions which give off or take in an electron, but the amount of energy gained or lost in such a reaction would be different in the two electrodes.  As a consequence, if neither electrode has a charge relative to the other, a combination of reactions will occur in the electrodes so as to build up a charge, until the amount of energy required to move an electron from one electrode to another [proportional to the voltage] equals the difference in the chemical energies between them.  If one connects an external resistance between the two electrodes, the number of electrons that would have to flow to maintain a constant voltage will equal the number of electrodes that are flowing through the external resistance.
