Does an electron move in a conductor? The definition of current is flow of electric charge. But recently I have heard that the electrons cannot move, that they just transmit energy to the other electrons and so on.
 A: Classically, electrons do move in a conductor that is passing direct current – but much more slowly than you might think. Let's break this down:
Current in a wire is defined as the amount of charge that passes through a cross-section of that wire in a single second. By this definition alone, it is clear that a current relies on the motion of some charged particle. It is possible that there could be a system where electrons transfer energy to each other, but in classical terms this would not be considered a "current." However, as I mentioned before, electrons actually move pretty slowly, even in very high-power currents. This might be what you're thinking of – how even very slow-moving electrons transfer a lot of power.
As a matter of interest, let's look at exactly how quickly electrons move. We need a common identity, $I = qnA\overline{v},$ where $q$ is the charge of the charge carrier, $n$ is the number of those particles per unit volume, $A$ is the cross-sectional area of the wire, and $\overline{v}$ is the average speed of these particles. This identity is fairly simple to derive – $q n$ is the charge density per unit volume, and $A \overline{v}$ is the average volume of particles that passes through a cross-section of the wire in a given second, so $q n A \overline{v}$ is the total charge that passes through a cross-section of a wire in a given second, or equivalently the current $I$.
Now, suppose we have a copper wire. Let's make it pretty thick – say, 1cm in diameter. Classically, a current in a copper wire is transmitted by electrons. So the charge of the charge carrier is $q = e = 1.6 \cdot 10^{-19}$ C. 
To find $n$, we note that, from Wikipedia, "Copper has a density of $8.94$ g/cm$^3$, and an atomic weight of $63.546$ g/mol, so there are $140685.5$ mol/m$^3$. In 1 mole of any element there are $6.02\cdot 10^{23}$ atoms (Avogadro's constant). Therefore in $1$ m$^3$ of copper there are about $8.5 \cdot 10^{28}$ atoms ($6.02 \cdot 10^{23} \cdot 140685.5$ mol/m$^3$). Copper has one free electron per atom, so $n$ is equal to $8.5 \cdot 10^{28}$ electrons per m$^3$."
For $A$, our wire is circular and has diameter $1 cm$, so its cross-sectional area in square meters is $A = \pi r^2 = \pi (0.5 \cdot 10^{-2})^2 = 7.85 \cdot 10^{-5}$ m$^2$
Now let's suppose we have a current of $1$ ampere – a fairly strong current by most standards. The velocity of the moving electrons in the wire is
$$
\overline{v} = \frac{I}{q n A} = \frac{1 \text{ C/s}}{1.6 \cdot 10^{-19} \text{ C} \cdot 8.5 \cdot 10^{28} \text{ m}^{-3} \cdot 7.85 \cdot 10^{-5} \text{ m}^2} \approx 9.37 \cdot 10^{-7} \text{m/s}.
$$
So a centimeter-thick copper wire carrying an ampere of current only requires its electrons to move $9.37 \cdot 10^{-7}$ m/s on average. That's very slow! You'll note from the relation $\overline{v} = I / q n A$ that the thicker the wire becomes, the smaller the velocity of the electrons is – that is, as $A$ grows larger, $\overline{v}$ grows smaller. Obviously, the wider a wire is, the more electrons can move through it. The more electrons moving through a wire, the slower they have to go to move the same total net charge.
The point is, power transferred in a wire is a result of massive numbers of electrons moving very, very slowly. They are moving, though. 
Of course, this all only holds for direct current – as some other users have mentioned, alternating current is another common current in which electrons move back and forth with some predetermined frequency. In this case, the current is constantly switching directions, hence the name "alternating." This also could be what you're thinking of – as the electrons are moving back and forth but staying in essentially the same place. It is not the individual electrons that are transferring energy to each other, however, but the electric field pervading the wire which is constantly switching direction and forcing the electrons in the wire to change their direction of motion.
A: When an electric field is applied on a conductor there exists a drift velocity for the electrons in the medium . Electrons in conductors exist in what is called the Fermi level, a  band  level common to the whole solid. Depending on the field they can accumulate on one side or the other leaving a positive charge from the combined field of the molecules on the opposite side.
When an alternating currentis applied they move back and forth with the applied frequency.
