Here is a simplified, but reasonably accurate description of what happens with electrons and electric fields. I say simplified because electromagnetic field theory has many "complications", describing which is the topic of many books.
It may be easier to start thinking about DC. If you look at a small enough time interval, AC looks like DC as the potential changes very little during that period of time, and can be considered as constant. The voltage source sets up an electric field in the wire by putting the opposite ends of the wire at different electric potentials (actually, the field is everywhere - it is just the conductor where the most interesting things happen). Voltage is the measure of the potential. The application of the potential can happen through various methods (magnetic, mechanical, or chemical), but isn't really what is important here - just that there is a potential difference applied.
This electric field created by the difference in potential exerts a force on the electrons, pulling them one way or the other. In some materials there are many "free" electrons that can move easily in large numbers ("conductors", like in copper wire), and in other materials there are few "free" electrons, and they are much more difficult to move in large numbers ("insulators", like most plastics and air). Electrons have a "charge" (which is actually what the electric field is applying the force on), and as charge moves, it is called "current". If there is a good conducting path from the area of high potential to low potential, the electrons, and hence charge, will move easily, creating current.
Now consider AC. What is happening is that the potential across the conductor is changing over time, so that at one point in time one end has a high potential and the other has a low potential, and later, they have switched so that the previously low potential end becomes high potential, and the previously high potential end becomes low potential. This causes the electrons to first move in one direction, and then to stop, and then move the other direction as the electric field changes direction. The actual distance the electrons travel in the conductor depends on the potential, how many electrons are available to move, how easily they move in the conductor, and the frequency with which the AC changes direction (the slower the change, the farther they go). With AC and typical lengths of conductors, it is unlikely that an electron from one end ever makes it all the way to the other end. But with DC or very short conductors, it is possible that electrons may travel all the way from one end to the other.
Another aspect is that when charge moves (current), it creates a magnetic field. It is basically this magnetic field that is measured by a clamp-on current meter. And prior to digital meters, it was actually this magnetic field that was used to measure both AC and DC currents and voltages (for voltage, a conducting path is provided so that current will flow in the meter). The magnetic field created by the current causes a force to be applied to a permanent magnet that is mounted on a needle, which generally prevented from moving by a spring. As the magnetic field increases due to an increase in current, it applies more force to the permanent magnet, pushing the needle against the force applied by the spring, causing it to move and display the current or voltage being measured.
