Width of a Confidence Interval vs Margin of Error

Width is what you trade against confidence: raising the confidence level raises the critical value and widens the interval, while a larger sample lowers the standard error and narrows it. Because the standard error carries a $\sqrt{n}$, the width is roughly proportional to $1/\sqrt{n}$, so cutting the width in half takes four times as much data. For example, the interval 12 to 18 has width $18 - 12 = 6$ and margin of error 3, and moving from n = 100 to n = 400 would shrink that width to about 3. A narrow interval only helps if the method is sound, since a biased sample gives you a tight interval around the wrong value.

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.