Electricity is not a substance that flows into a wire from outside. It is the collective behaviour of a property that matter already carries everywhere, all the time, called charge. A circuit is charge being pushed around a loop, resistance is charge being slowed, a capacitor is charge being stored, and alternating current is charge being rocked back and forth. Before any of that can make sense, the property itself has to be pinned down, and it turns out to have just a few rules, from which the whole subject is built.
Two kinds, and the sign convention
Charge comes in two kinds, and the two are opposite in a strict sense: bring like kinds together and they push apart, bring opposite kinds together and they pull in. Benjamin Franklin named them positive and negative before anyone knew what carried them, and the names stuck, which is why the particle we now know does most of the moving in a wire, the electron, is negative. That was a coin flip Franklin lost, and it leaves the field with a permanent awkwardness: the thing that physically flows in a metal, the electron, moves opposite to the direction we call the current. The subject lives with the confusion by keeping two ideas apart, and this lesson keeps them apart too.
Charge is measured in coulombs, symbol C, and the coulomb is a large unit: the charge on a single electron is only C, so one coulomb is the combined charge of about six billion billion electrons. Two more facts complete the rules. Charge is quantised, always a whole-number multiple of that electron charge, never a fraction of it. And charge is conserved: it is never created or destroyed, only moved, so any charge that leaves one place has gone to another. Conservation is not a detail. It is the reason current is the same all the way around a simple loop, a fact the whole of circuit analysis leans on.
Coulomb's law, the force between charges
Two charges exert a force on each other, and its size follows a law with the same shape as gravity's. The force between charges and separated by a distance is
where in units of newton metres squared per coulomb squared. The force falls off as the square of the distance, so doubling the separation quarters the force, and it grows with the product of the charges. Its direction is along the line joining them, a push for like charges and a pull for opposite ones, which the sign of the product encodes automatically. This inverse-square law is the whole of electrostatics in one line, and every richer idea in the subject is a way of applying it to more charges than two.
Example. Two small spheres each carry C and sit 5 cm apart. What is the force between them, and which way does it point?
N, and since both are positive the force is a push, driving the spheres apart. Fourteen newtons from two specks of charge is a hint of how strong the electric force is compared with gravity.
Now you. The spheres are moved to 10 cm apart, charges unchanged. What is the new force?
Answer
Doubling the distance quarters the force, so N, still a push.
The field, action carried through space
Coulomb's law gives the force between two charges, but it is awkward to think of every charge reaching out and touching every other one directly. The more powerful idea, and the one the rest of the subject uses, is that a charge fills the space around it with an electric field, and any other charge placed in that space feels a force from the local field rather than from the distant charge. The field at a point is the force per unit charge that a small positive test charge would feel there:
measured in newtons per coulomb, or equivalently volts per metre once the next lesson introduces the volt. The field points away from positive charge and toward negative charge, and its strength falls off with distance just as the force does. The value of the field idea is that it lets a charge respond to what is happening right where it sits, with no need to know where the charges that made the field are. In a wire, the field is what pushes the charges along, set up by the battery at the ends, and every charge in the wire feels the local push and moves, which is the beginning of a current.
Conductors and insulators
Matter divides, for this subject, into two camps by what its charges are free to do. In a conductor, above all a metal, some electrons are not bound to any one atom but are free to roam through the whole material, so an applied field sets them drifting and charge flows. In an insulator, glass, plastic, dry air, every electron is held tightly to its atom, so a field shifts them only slightly and no sustained flow occurs. This single difference is why a circuit is built of metal wires wrapped in plastic: the metal offers the charge a road, and the plastic walls it in so the charge goes where it is wanted and not through the person holding it. The free electrons in a metal are the movable charge of every lesson that follows, and the field that pushes them, set up by a source with the strange power to keep pushing, is the subject of the next lesson: what a voltage actually is.