For example, for an i.i.d. sample of random variables $X_i$ distributed according to a normal distribution, I found a sufficient statistic—the sample mean. How do I know if this is also complete? Thank you.
I am sure you know the formal definition of completeness, so I will try to offer some intuition on the subject.
Informally a statistic $T(X)$ is complete if two different parameters $\theta$ of the distribution of X, cannot give rise to the same distribution for $T(X)$.
The distribution of sample mean $\bar X$ is $N(\mu,\sigma^2/n)$, which is different if you change either $\mu$ or $\sigma^2$. So by the above intuition, it is complete.
I hope that helps. Examples above are assuming that the variance $\sigma^2$ is also a parameter.