Circuits diagrams are historically simply diagrams of actual electronic circuits. From the diagram, you can put together an electronic device. However, they have become an abstract language, with a set of mathematical rules

Quantities
Charge (Q) (Measured in Coulombs)
This is the "thing" that flows around the circuits. It is a conserved quantity; in any electric process it is neither created or destroyed. Although many of the so called 'elementary particles' have electric charge, in electronics you worry only about electrons. Because each electron has a fixed charge of -1.6E-19 Coulomb, you can interpret charge as the number of electrons present. The 'zero'-point in not zero electrons, but the amount of electrons required to make an object electrically neutral. (=Have zero net charge) Ordinary matter contains many protons, who are positively charged. They do not participate in electricity, except that they balance the charge of the negatively charged electrons.

Current (I) Measured in Ampere
The amount of charge flowing through a component per second

Mesh current (J). This is the current that goes around in a loop of a circuit. The other currents can be expressed as sums of mesh currents, and vice versa.

Potential (V) , often called Voltage, as it is measured in Volts
The energy in Joule per coulomb that a charge has. Strictly speaking you can only define differences in potential,  no absolute potential. This is especially true in the diagram for  Maxwell equations  .
The energy is related to the fact that electrons repel each other. They exert a force on each other, and forces are related to energy. But in electronics, you don't worry about individual electrons, you just say that the charge has a certain energy at a certain position. As it flows to point with a different potential, the energy is converted into other forms. This can be heat,  magnetic energy, or electrostatic energy. Electrostatic energy is electric energy stored in the vacuum. (In our article the vacuum is imagined to be filled with capacitors, which contain the electrostatic energy.)

Note that this definition is only valid for infinitesimal amounts of charge. For larger charges, moving the charges will influence the fields. For example, the energy in a capacitor is:

    E=(1/2)Q^2/C

But the potential is:

    V=dE/dQ = Q/C

This is twice the average energy of the charge in the capacitor.
 

Power (P) Measured in Watts
You get the power converted by a component when you multiply the current with the potential difference across the component.

Concepts

Complex impedances

****Under construction*****
 

Electric 2-poles

Resistance, Inductors, an capacitors are 2- poles. That means that they are objects that are idealized to have only two interfaces with the outside world, the poles. This abstraction allows you to define the properties of the objects in terms of voltage difference between the poles, and current going from pole to pole. Because of charge conservation the charge that goes in also comes out, unless it is stored. The only 2-pole we use that can store charge is the capacitor. Otherwise, there is just one current for each 2-pole. But even for a capacitor, you usually say there is only one (complex) current.
 
Components
Resistance (R) Measured in Ohm

A resistor is a special case of a 2-pole.

When electricity passes through matter, it generates heat. The energy of this heat is drawn from the electric energy, so there must be a potential across the resistance. The resistance is defined as:

    R=Delta(V)/I

This is also known as Ohm's law.  It is often written as
 
    V=I*R.

V should really be Delta(V), but we are careless.

Ohm's law can be generalized for complex impedances. It then applies not only for resistors, but also to inductors and capacitors. The symbol (R) for resistance is then replaces by (Z) for impedance
 
   Delta(V)=I *Z

Usually, the value of the resistance in constant, it does not depend on V or I. Given the resistance you can use Ohm's law to calculate Voltages in term of currents and vice versa.

The symbol is:

Capacitance (C), measured in Farad.

A Capacitance is an object that stores electric charge. It is a 2-pole.  The  storage of charge gives the charge energy, analogous to a container with water (=charge), being filled, and getting energy from the height of the water (h) in the gravitational field (g). The energy of a small piece of water dm is

dE = dm*g*h

The height is related to the total mass of water (m) by

    h = m / A

where A is the Surface area of the container.
It follows that the total energy is:

    E = integral(dE) = integral(dm*g*h) =  integral(dm*g*m / A) = (1/2) m^2*g/A
 
In analogy, the energy in the capacitor is:

    E=(1/2)*Q^2/C

 The voltage across the capacitor is:

    V=Q/C

or:

    dV/dt = I/C

or, using complex impedances

    V=I/(jwC)

So, the impedance of the capacitor is:

    Z=1/(jwC)

It becomes infinite as w goes to zero. This corresponds to the fact that if you continue charging it indefinitely, the Voltage goes up indefinitely.

 
 
Inductance (L) measured in Henri

An inductance is mathematically similar to the capacitor, you just change the Voltage and the current around:

    dI/dt = I/L

or, using complex impedances

    V=I*(jwL)

So, the impedance of the inductor is:

    Z=jwL
 
 

An inductor appears to resist change in current, because the current is coupled to magnetic energy. If you change the current, you have to change the magnetic energy, and you must do work.
 
The energy is, in analogous to the capacitor :

    E= (1/2)*L I^2

Kirchhoff's current law.
It follows from the conservation of charge that the sum of currents into a node is zero. This is Kirchhoff's current law.
There is also Kirchhoff's voltage law, which is says that the sum of voltage differences around a loop is zero. In the case of a changing magnetic flux through this loop , you have to add the induction voltage to this.