Precision
Precision is how closely repeated measurements agree with each other, reflecting the fineness of the instrument and the size of random error.
Definition
Precision is how close your repeated measurements are to each other: measure the same thing three times, get almost the same number each time, and your measurements are precise. It is the degree of agreement among repeated measurements, shown by a small range or standard deviation, and it reflects the instrument's resolution and the amount of random error in the procedure; a finer instrument lets you report more significant figures, and a better procedure makes trials cluster more tightly. Precision quantifies random error and is improved by finer instruments, better technique, controlled conditions, and more trials, since the uncertainty of the mean falls as $1/\sqrt{n}$, but no amount of precision compensates for systematic error. Instrument precision (resolution) and procedural precision (repeatability) are distinct, and reported significant figures should reflect the poorer of the two.
Example
Three cart runs of $84$, $85$, and $84\ \text{cm}$ are precise; runs of $70$, $95$, and $82\ \text{cm}$ are not, even if the average happens to be right. A cylinder marked every $1\ \text{mL}$ gives readings like $23.4$, $23.5$, and $23.4\ \text{mL}$ with a range of $0.1\ \text{mL}$, far more precise than a beaker marked every $50\ \text{mL}$. Five density trials of $2.68$, $2.71$, $2.69$, $2.72$, and $2.70\ \text{g/cm}^3$ have a mean of $2.70$ and a standard deviation of about $0.016\ \text{g/cm}^3$, a relative precision near $0.6\%$ that is limited by the $\pm 0.1\ \text{mL}$ volume reading rather than the $\pm 0.01\ \text{g}$ balance.
Key Insight
Precise means tightly grouped, like darts landing close together; it does not say whether they landed near the bullseye. Precision is not accuracy, and a miscalibrated balance can give $12.47\ \text{g}$ every single time while being wrong by half a gram. Improving precision is worthwhile only until random error is smaller than the systematic error; beyond that, effort belongs in calibration, not in more decimal places.