Given data (vector) $y$ and unknown parameter (vector) $\theta$, the posterior distribution is given by
$$
p(\theta|y) = \frac{p(y|\theta)\,p(\theta)}{p(y)} ,
$$
where $p(y|\theta)$ is the likelihood, $p(\theta)$ is the prior, and $p(y)$ is (often called) the marginal likelihood.

The Metropolis-Hastings algorithm can be summarized as follows. Let $q(\theta'|\theta)$ denote the density for the proposal distribution for $\theta'$ conditioned on $\theta$. Let $\theta^{(r)}$ denote the current state of the MCMC chain. Then
\begin{equation}
\theta^{(r+1)} = \begin{cases}
\theta' & \mathcal{R} \ge u \\
\theta^{(r)} & \text{otherwise}
\end{cases} ,
\end{equation}
where $u \sim \textsf{Uniform}(0,1)$ and
\begin{equation}
\mathcal{R} = \frac{p(\theta'|y)}{p(\theta^{(r)}|y)} \times \frac{q(\theta^{(r)}|\theta')}{q(\theta'|\theta^{(r)})} .
\end{equation}
Note that $p(y)$ cancels out of the first factor on the right-hand side. Also note that if
\begin{equation}
q(\theta|\theta') = q(\theta'|\theta)  \qquad\text{for all $(\theta,\theta')$}
\end{equation}
then the second factor disappears and we have the Metropolis algorithm.