Electricity is "flow of electrons". My child asked me if this is so, then ultimately the copper wire should disappear/vanish/finish because the matter is moving from one place to the other. I am not electrical engineer, what should I tell him?
In a metal like copper some of the electrons are not bound to individual atoms. If a voltage is applied across a copper wire these free electrons flow from one atom to the next. This flow of electrons is an electric current but the copper atoms themselves do not move so the copper wire does not vanish. Of course, the voltage source must supply additional electrons for the flow to continue.
As an analogy, think of a waterfall -- the electric current is like the water molecules falling, and the water molecules are falling due to gravity (which is analogous to voltage). The waterfall must be supplied with more water molecules for the waterfall to continue to exist, but the riverbed over which the water molecules flow (analogous to the copper atoms) do not move or vanish.
To add another 2 cents, and a rather more simplistic understanding, I often find it easier to visualise current flow through a wire as tube of marbles (small glass balls if that doesn't translate well).
The wire has electrons in it already, so our tube is full of marbles. By applying a voltage (an electromotive force), you can push a new marble in. When you do this, a marble pops out. Take the one that popped out, and push it in the other end. In a real circuit, it doesn't have ends to push electrons in and out, but it flows all the way around (so our tube would be joined together at both ends).
In electronics, we don't create or destroy, add or remove electrons* - they are already there. All we do is push them along. It's also important to understand that it isn't one electron be pushed around, it's the flow. You push one end, and the other end moves. Pushing electrons at one end doesn't mean the same electron pushes at the other end, just like pushing marbles a different one comes out the end.
As you delve further and further down, you can begin to appreciate holes and carriers and electrons jumping up and down energy levels; but it simplistic terms, just imagine pushing marbles in a tube.
* Yes there are probably exceptions, but they aren't relevant.
The simplest answer you can give, is that for every electron that comes out of one end of the wire, another electron is "pushed" in at the other end of the wire. Therefore, even though there is a "flow" of electrons, the wire does not loose any electrons (no net loss)!
I am an electrical engineer. And of course I agree with the answers above that electrons flow both into and out of the wire. But it is interesting to see an example of how slow that flow really is for normal currents. For simplicity, let's make this a direct current example.
Let's imagine a length of 10 gauge copper wire (which is roughly 0.1 inch in diameter), that has a 10 ampere current flowing through it (fairly typical household number). Further, let's assume that all the "free electrons" from the copper atoms (that's 1 per atom) are participating in the flow of current. This means that the speed of the flowing electrons is about 1 INCH EVERY 2.5 MINUTES. Slower than most of us would guess.