Margin of Error vs Standard Error

The margin of error sets the reach on each side of a point estimate in a confidence interval, so the interval is $\text{estimate} \pm \text{margin of error}$. It grows with the confidence level and shrinks as the sample size grows, since more data lower the standard error. For example, a poll reporting 47% with a 3% margin of error gives the interval 44% to 50%. In symbols, $\text{margin of error} = z^* \cdot \frac{\sigma}{\sqrt{n}}$ for a mean with known population standard deviation $\sigma$, where $z^*$ is the critical value and $n$ the sample size.

The standard error plays the same role for a statistic that the standard deviation plays for raw data, and it gets smaller as the sample size grows. For example, the standard error of a sample proportion is $\sqrt{\frac{p(1-p)}{n}}$ (the square root of p times one minus p, divided by n). For a sample mean it is $\frac{\sigma}{\sqrt{n}}$ (sigma, the population standard deviation, over the square root of n). A smaller standard error means the estimate is more precise, which narrows a confidence interval.