Is this interpretation of the EM algorithm correct? Suppose we want to estimate the parameter $\theta$ of the distribution $p(z;\theta)$ of a random variable $Z$. If we had the i.i.d samples $z_1, z_2, \dots, z_N$, we could easily do this using maximum likelihood estimation. However, suppose we are only given the samples $x_1,x_2,\dots,x_N$ of another random variable $X$, where $X$ and $Z$ are related via the joint distribution $p(x,z)$. Since $p(x,z) = p(x \mid z) \cdot p(z;\theta)$, then $p(x,z) = p(x,z;\theta)$.
The objective of the EM algorithm is to estimate the samples $z_1, z_2, \dots, z_N$ using the given samples $x_1,x_2,\dots,x_N$, and then use these estimated samples to perform maximum likelihood estimation to estimate $\theta$. However, it turns out that estimating the samples $z_1, z_2, \dots, z_N$ using the samples $x_1,x_2,\dots,x_N$ requires knowledge of $\theta$, which is what we are trying to estimate in the first place. Therefore, we initialize a value for $\theta$, estimate the samples $z_1, z_2, \dots, z_N$, then estimate a new value for $\theta$, and repeat the process until convergence.
More formally, if
$$
p(z_1, z_2, \dots, z_N;\theta)
$$
is the likelihood function for $\theta$, then to maximize this likelihood function, we need to estimate the samples $z_1, z_2, \dots, z_N$ as $\hat{z}_1, \hat{z}_2, \dots, \hat{z}_N$ such that
$$
\hat{z}_i = \mathbb{E}[Z \mid x_i] \quad \forall i \in \{1,2,\dots,N\}
$$
We could then plug these estimates into the likelihood function,
$$
p(\hat{z}_1, \hat{z}_2, \dots, \hat{z}_N;\theta)
$$
to determine $\theta$. However, as each $\hat{z}_i$ depends on $\theta$, we need to use the iterative process mentioned above.
Is this interpretation correct?

This is kind of half right. Obviously the main objective is to estimate $\theta$ from $x$. Given knowledge of $z$, we can maximise the likelihood derived from $p(z;\theta)$. This is done in the M-step. What is computed in the E-step is the expected value of that likelihood given $x$ and the $\theta$ estimated in the earlier M-step (or, in the beginning, its initial value). This amounts in some sense to the "estimation" of $z_1,\ldots,z_N$, as $x$ is known and $z$ is treated as random in $p(z;\theta)$. However, in most cases it isn't really estimating the individual observations as $\hat z_1,\ldots,\hat z_N$ and then plugging them in. For example, in mixture models, where $Z=(X,Y)$ and $Y$ is the mixture component to which an observation belongs (out of components $\{1,\ldots,K\}$, say), what is done is that the probabilities $P\{Y_i=k|x_i;\theta\}$ are estimated, i.e., a vector of $K$ conditional probabilities, rather than just a single $\hat y_i$ (or $\hat z_i$ by implication). Ultimately it depends on the form of the likelihood of $p(z;\theta)$ what exactly is done.
