The effective resistance begs the question: “What single resistor can be used to replace all of these other resistors in my circuit?”. We already know how voltage and current behave in series and parallel circuits so we can easily deduce how the resistance, by Ohm’s Law, will behave.
Effective/equivalent resistance of resistors in series
According to Kirchhoff’s Voltage Law, voltage is additive along a loop (yes, for resistors in series): $$ \begin{equation}\begin{aligned} V_T=V_1+V_2+V_3\\ \end{aligned}\end{equation} $$ Current is also constant due to the Principle of Conservation of Charge: $$ \begin{equation}\begin{aligned} I_T=I_1=I_2=I_3\\ \end{aligned}\end{equation} $$ Dividing each voltage by its corresponding currents (this is allowed because current is the same): $$ \begin{equation}\begin{aligned} \frac{V_T}{I_T}&=\frac{V_1}{I_1}+\frac{V_2}{I_2}+\frac{V_3}{I_3}\\ \end{aligned}\end{equation} $$ Because $\frac{V}{I}=R$: $$ \begin{equation}\begin{aligned} \therefore R_T&=R_1+R_2+R_3\\ \end{aligned}\end{equation} $$
The effective resistance of resistors arranged in series is the sum of the individual resistances.
Effective/equivalent resistance of resistors in parallel
By KCL, the current flowing through resistors in parallel is additive: $$ \begin{equation}\begin{aligned} I_T=I_1+I_2+I_3\\ \end{aligned}\end{equation} $$ The voltage drop across each resistor is the same: $$ \begin{equation}\begin{aligned} V_T=V_1=V_2=V_3\\ \end{aligned}\end{equation} $$
Dividing the formula for current throughout by the voltage: $$ \begin{equation}\begin{aligned} \frac{I_T}{V_T}&=\frac{I_1}{V_1}+\frac{I_2}{V_2}+\frac{I_3}{V_3}\\ \end{aligned}\end{equation} $$ Because $\frac{V}{I}=\frac{1}{R}$: $$ \begin{equation}\begin{aligned} \therefore \frac{1}{R_T}&=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\\ \end{aligned}\end{equation} $$
The reciprocal of the effective resistance of resistors arranged in parallel is the sum of the reciprocals of the individual resistances.