The standard error of an estimator is the standard deviation of its sampling distribution.
It measures how much of the estimator would vary across repeated random samples from the population. A smaller standard error means the estimate is more precise. For the sample mean, the standard error decreases as $1/\sqrt n$ under independent observations with a common finite variance. Thus, increasing the sample size reduces uncertainty, but with diminishing returns.
For an estimator $\hat\theta$, the standard error is the square root of the variance:
$$ \text{SE}(\hat\theta)=\sqrt{\text{Var}(\hat\theta)} $$
For $n$ independent observations $X_1, \ldots, X_n$ with population standard deviation $\sigma$ the sample mean:
$$ \bar X=\frac{1}{n}\sum_{i=1}^{n}X_i $$
has standard error:
$$ \text{SE}(\bar X)=\frac{\sigma}{\sqrt{n}} $$
Because the standard deviation $\sigma$ is usually unknown, it is estimated using the sample standard deviation $s$:
$$ \widehat{\text{SE}}(\bar X)=\frac{s}{\sqrt{n}} $$