The Black-Scholes equation is $$ \frac{\partial C}{\partial t} + \frac{\partial C}{\partial S} r S + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 - rC = 0 $$ where
$t$ is the current time
$S$ is the spot price of the asset
$C$ is the option price
$r$ is the risk-free interest rate
$\sigma$ is the volatility parameter of the underlying. This partial differential equation describes the evolution of the price of a call option as a function of the current spot price of the underlying and the time to expiry. There are several assumptions that underpin the Black-Scholes equation. For example, we assume that the evolution of the stock price is determined by the stochastic differential equation $$ \text{d}S = rS\text{d}t + \sigma S \text{d}W. $$