Electrical power, $P$ can be defined as the rate at which electrical energy $\left(W\right)$ is consumed in an electrical circuit.
$\Rightarrow$$P=\frac{W}{t}$
 Potential difference, $V$ across any circuit is defined as the electrical energy $W$ supplied by a source for a unit flow of charge ($I\times t$) 
$\therefore V=\frac{W}{I\times t}$ 
$\Rightarrow W=V\times I\times t$

So, power, $P=\frac{V\times I\times t}{t}=VI$

By Ohm's law, $V=IR$ or $I=\frac{V}{R}$

Substituting the value of $I$ in the expression of power, $P=VI=V\times \frac{V}{R}=\frac{V^2}{R}$

Since, $V=IR,P=\frac{V^2}{R}=\frac{(IR{)}^2}{R}=I^{2}R$

Thus, electrical power $P$ dissipated in a resistor $=\frac{W}{t}=VI=\frac{V^2}{R}=I^{2}R$.