The connection between both concepts is that Markov chain Monte Carlo
(aka MCMC) methods rely on Markov chain theory to produce
simulations and Monte Carlo approximations from a complex target
distribution $\pi$.
In practice, these simulation methods output a sequence $X_1,\ldots,X_N$ that is a Markov chain, i.e., such that the distribution of $X_i$ given the whole past $\{X_{i-1},\ldots,X_1\}$ only depends on $X_{i-1}$. In other words, $$X_i=f(X_{i-1},\epsilon_i)$$ where $f$ is a function specified by the algorithm and the target distribution $\pi$ and the $\epsilon_i$'s are iid. The (ergodic) theory guarantees that $X_i$ converges (in distribution) to $\pi$ as $i$ gets to $\infty$.
The easiest example of an MCMC algorithm is the slice sampler: at iteration i of this algorithm, do
- simulate $\epsilon^1_i\sim\mathrm{U}(0,1)$
- simulate $X_{i}\sim\mathrm{U}(\{x;\pi(x)\ge\epsilon^1_i\pi(X_{i-1})\})$ (which
amounts to generating a second independent $\epsilon^2_i$)
For instance, if the target distribution is a normal $\mathrm{N}(0,1)$ [for which you obviously would not need MCMC in practice, this is a toy example!] the above translates as
- simulate $\epsilon^1_i\sim\mathrm{U}(0,1)$
- simulate $X_{i}\sim\mathrm{U}(\{x;x^2\le-2\log(\sqrt{2\pi}\epsilon^1_i\})$,
i.e., $X_i=\pm
\epsilon_i^2\{-2\log(\sqrt{2\pi}\epsilon^1_i)\varphi(X_{i-1})\}^{1/2}$
with $\epsilon_i^2\sim\mathrm{U}(0,1)$
or in R
T=1e4
x=y=runif(T) #random initial value
for (t in 2:T){
epsilon=runif(2)#uniform white noise
y[t]=epsilon[1]*dnorm(x[t-1])#vertical move
x[t]=sample(c(-1,1),1)*epsilon[2]*sqrt(-2*#Markov move from
log(sqrt(2*pi)*y[t]))}#x[t-1] to x[t]
Here is a representation of the output, showing the right fit to the $\mathrm{N}(0,1)$ target and the evolution of the Markov chain $(X_i)$.

And here is a zoom on the evolution of the Markov chain $(X_i,\epsilon^1_i\pi(X_i))$ over the last 100 iterations, obtained by
curve(dnorm,-3,3,lwd=2,col="sienna",ylab="")
for (t in (T-100):T){
lines(rep(x[t-1],2),c(y[t-1],y[t]),col="steelblue");
lines(x[(t-1):t],rep(y[t],2),col="steelblue")}
that follows vertical and horizontal moves of the Markov chain under the target density curve.