This representation of the Black Scholes model consists of a two asset market. The first is one risky asset $S_t$ such as a stock, future forward or foreign currency. As before this risky asset (the underlying) follows a geometric Brownian motion given by the stochastic differential equation $$dS_t = \mu S_t dt + \sigma S_t dW_t \tag{1}$$ where $(W)_{t\geq 0}$ is a $\mathbb{P}$ Brownian motion, $\mu$ and $\sigma$ are constants.
The source of risk is the $W_t$ and hence all risk is captured by the risky asset $S_t$. The GBM price of $S_t$ is given as $$S_t = S_0 e^{(\mu – \sigma^2/2)t – \sigma W_t}$$
The second asset is a risk-less money market account $M_t$ such as a bond, a savings account that evolves according to $$dM_t = rM_t dt \tag{2}$$ This gives $$\dfrac{dM_t}{dt} = rM_t$$ This has solution $$M_t = M_0 e^{rt}$$ which if normalised by setting $M_0 = 1$ becomes $$M_t = e^{rt}$$ Note: Some texts use the notation $B_t$ for the money market account $M_t$, other texts use $S_0$.
The goal of martingale pricing is to find a self-financing strategy to replicate a European option payoff at $T$. We must make the usual simplifying assumptions in our Black and Scholes Model.