There are two main ways in which electronic components are connected to each other to form useful circuits.
In the series circuit, the components are connected one after the other forming a line. What this configuration does it that it gives only path for current to flow through the circuit. One of the properties of series circuit is that the current is the same in all parts of the circuit.
If connecting a number of resistors in series, there's an equivalent resistor that represents the total resistance from the point of view of the source.
That happens because of the second Kirchhoff law. The voltage supplied is the same as the voltage induced in the resistors.
Let's use an example circuit with one voltage source and three resistors in series.
Expressing it mathematically, we have that
Vt = Vr1 + Vr2 + Vr3
where Vt is the total voltage supplied by the source, Vr1 is the voltage induced in resistor one and so on.
We also have that
V = I R, defined by ohm's law;
We also know that I is the same in all parts of the circuit, so
Ir1 = Ir2 = Ir3 = It;
So you can now change the first equation to
It Rt = [It R1] + [It R2] + [It R3]
Where Rt is the equivalent resistance of the circuit.
You can see that all values are multiplied by It, so to simplify, you can divide on both sided by It:
Rt = R1 + R2 + R3
Now you end up with an equation that defines the Equivalent resistance that the voltage source sees. Since the equivalent resistance of the circuit is the sum of all individual resistances, it will always be larger than any individual resistor, and by ohm's law, the current flowing through it will also be lower than through any individual component.
The special case of two series resistors is called a voltage divider, and is one of the simplest yet useful subcircuits you will find in many electronic circuits.
Since the voltage across the resistors depends on the current flowing through them, adding more circuitry to the output of the divider will draw more current. This extra current induces a different voltage than if the circuit was made of just the two resistors, so the output voltage changes accordingly to satisfy kirchoff's voltage laws.
This effect of change in operation in a circuit when more components are added is called loading.
Adding branches: Parallel circuits
When more there is more than one way for current to flow, the branches of the circuit are said to be connected in parallel.
In a parallel circuit, every branch of the circuit gets the same voltage, because they are connected to the exact same points.
Let's demonstrate it using the same process as with series circuits.
Imagine you have a circuit with one voltage source and three resistors connected in parallel. As you can see, all three resistors are connected across the voltage source, so they all get the full of the voltage induced on them.
You can visualize this in terms of kirchoff's laws if you make loops withing the circuit. One restriction is that the line that makes the loop can't pass twice through the same spot. If you make a loop with the voltage sources and any of the resistors, you can see that the voltage induced in the resistor will be the same as the voltage provided by the source in order to comply with the circuit law.
Now, to know the equivalent resistance that the voltage source sees, let's express the circuit in mathematical terms.
Since voltage does not change in any of the loops, we will use the total Current drawn by the circuit as a starting point.
It = Ir1 + Ir2 + Ir3
Where It is the total current drawn by the circuit, Ir1 is the current drawn by R1 and so on.
You also know that
I = V / R
You can now change the equation in terms of the known voltage and the known resistors
Vt / Rt = [Vt / R1] + [Vt / R2] + [Vt / R3]
If you divide both sides by Vt you get
1 / Rt = + [1 / R1] + [1 / R3] + [1 / R3]
This doesn't make much sense as it is, because we still need to get the inverse of both sides of the equation to get to the equivalent resistance. This equation, as it is, makes far more sense when you know another characteristic of conducting materials: conductance
