Estimators are random variables. They exhibit properties that we use to assess their quality, advantages, and disadvantages. So it depends what you mean by "is an estimate of." I can say $\hat{\mu}_0 = 0$ is an estimate of $\mu$, but that doesn't mean it's useful or successful (it uses absolutely no information).
The three most popular properties of statistical estimators in my experience are the following:
- Bias
- Efficiency
- Consistency
The estimator I proposed $\hat{\mu}_0$ is a biased, inconsistent, and inefficient (i.e. it just sucks) estimator of $\mu$. When the $X_i$ are drawn iid from the population distribution and when the expectation of the population distribution exists (eg. Cauchy distribution does not satisfy this latter assumption) and is finite, the arithmetic mean estimator
$$\bar{X} := \frac{1}{n}\sum_i X_i$$
is an unbiased and consistent estimator of $\mu$. Efficiency is a matter of relativity, but Cramér and Rao have shown that when the population distribution is Gaussian or Bernoulli, the arithmetic mean is (asymptotically) the most statistically efficient estimator. (I think there are few more weak regulation conditions needed to establish that result, but that's the main idea.)
All in all, $\bar{X}$ is a great estimator. But when data is corrupted by outliers or when the population distribution has heavy tails (two things that aren't true of Gaussian or Bernoulli distributions), this thing can be somewhat unstable. That's when the tradeoff between the properties come to play: Can we design a new estimator that sacrifices a little bias for a huge advantage in efficiency? What is required of the outliers to ensure that $\bar{X}$ remains consistent and is this a realistic assumption? And so on, and so on.