What is the Difference between a Series and Parallel Connection of Resistors?
Answer #1
Video  The 5 Differences Between Parallel and Series Circuit
This is how series and parallel circuit with three resistances \(R_1\), \(R_2\) and \(R_3\) looks like:
You can see the first difference in the circuits and namely how the resistors are connected together:

In a series circuit, the positive pole of the voltage source that supplies the voltage \(U\) is connected to a single resistor \(R_1\) and the negative pole is also connected to a single resistor, namely \(R_3\). The resistors in a series circuit form a chain to which you apply a voltage \(U\).

In a parallel circuit, on the other hand, you connect the positive pole of the voltage source to three ends of the three resistors, and you do the same with the negative pole.
Furthermore, a parallel circuit differs from a series circuit of resistors in the following points:

In a series circuit, the same current \(I\) flows through all three resistors:
Formula: Total current is equal to the individual currents in a series circuitFormula anchor $$ \begin{align} I ~=~ I_1 ~=~ I_2 ~=~ I_3 \end{align} $$In a parallel circuit, on the other hand, different currents \(I_1\), \(I_2\), and \(I_3\) flow through the resistors. The total current is the sum of the individual currents:
Formula: Total current is equal to the sum of the individual currents in a parallel circuitFormula anchor $$ \begin{align} I ~=~ I_1 ~+~ I_2 ~+~ I_3 \end{align} $$ 
In a series circuit, a different voltage \(U_1\), \(U_2\) and \(U_3\) is applied to each resistor. The total voltage \(U\) is the sum of the individual voltages:
Formula: Total voltage is the sum of the individual voltagesFormula anchor $$ \begin{align} U ~=~ U_1 ~+~ U_2 ~+~ U_3 \end{align} $$In a parallel circuit, on the other hand, the same voltage \(U\) is applied to all three resistors:
Formula: Total voltage is equal to the individual voltagesFormula anchor $$ \begin{align} U ~=~ U_1 ~=~ U_2 ~=~ U_3 \end{align} $$ 
The total resistance \(R\) of a series circuit is obtained by adding all individual resistances:
Formula: Total resistance is equal to the sum of the individual resistancesFormula anchor $$ \begin{align} R ~=~ R_1 ~+~ R_2 ~+~ R_3 \end{align} $$In a parallel circuit, however, the total resistance is not the sum of the individual resistances:
Formula: The reciprocal of the total resistance is equal to the sum of the reciprocals of the individual resistances.Formula anchor $$ \begin{align} \frac{1}{R} ~=~ \frac{1}{R_1} ~+~ \frac{1}{R_2} ~+~ \frac{1}{R_3} \end{align} $$