Accuracy refers to the closeness of a measured value to a standard or known value. For example, if in lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or known weight is 10 kg, then your measurement is not accurate. In this case, your measurement is not close to the known value.
Precision refers to the closeness of two or more measurements to each other. Using the example above, if you weigh a given substance five times, and get 3.2 kg each time, then your measurement is very precise. Precision is independent of accuracy. You can be very precise but inaccurate, as described above. You can also be accurate but imprecise.
For example, if on average, your measurements for a given substance are close to the known value, but the measurements are far from each other, then you have accuracy without precision.
A good analogy for understanding accuracy and precision is to imagine a basketball player shooting baskets.
If the player shoots with accuracy, his aim will always take the ball close to or into the basket. If the player shoots with precision, his aim will always take the ball to the same location which may or may not be close to the basket. A good player will be both accurate and precise by shooting the ball the same way each time and each time making it in the basket.
As far as widespread use goes, accurate and precise mean pretty much the same thing. It's when they're used scientifically that their definitions diverge.
Accuracy (for accurate) is defined as:
>the condition or quality of being true, correct, or exact: freedom from error or defect; precision or exactness; correctness.
A form of the word precise is even in the definition. However, when it comes to science, accurate/accuracy deals specifically with measurements thusly:
>the extent to which a given measurement agrees with the standard value for that measurement.
In other words, if something has a certain measure, and that's the answer drawn through measuring, experimentation, etc., the resulting measurement is said to be accurate.
For precision (or precise) we have:
>the state or quality of being precise.
>mechanical or scientific exactness.
So, again, we see that the other term, accuracy, pops up in the definition.
However, the scientific use changes a bit in that:
>precision refers to the closeness of two or more measurements to each other.
So, instead of measuring against a scientific standard, the measurements are against one another. If they come out the same each time, they are considered precise.