The concept of uncertainty in measurement has been documented as far back as 1000 AD with the Trial of Pyx at which it was all minted coins in Britain had to meet specific standards. In particular, it stated that a particular coin had an allowance of 5 grains^{1.} (0.324 grams). Now many scientists have stated that nonnumerical knowlege is unimportant; Lord Kelvin went so far as to label it "meagre and unsatisfactory." This view was shared by many scientists including Leonardo DaVinci and Francis Bacon.
In terms of science, however, uncertainty simply states how accurately a particular measuring tool is. Sometimes, uncertainty is stated in the form:
0.500 m ± 0.005 m
So for example a piece of wood is measured it would be exactly 50 cm long with an uncertainty of ± 0.005 m or 0.5 millimeters. this means that the measuring tool (for example, a meter stick can not be read any more accurately than one half a millimeter. Now consider the Swipe Ruler to the left:
A millimeter is the distance between the two smallest lines on the lower side of the ruler. It is difficult to get an accurate measurement between those two lines, so it is safest to say that the accuracy of the tool is 0.5 millimeters.
The other interesting thing about uncertainty is that a measurement can only be one decimal point beyond the smallest physical measurement. This means that for the case of a millimeter, the only possible smallest measurements are 0.1 millimeter ( 0.0, 0.1, 0.2, etc), 0.2 millimeters (0.0, 0.2, 0.4, etc.) and 0.5 millimeters ( 0.0, 0.5, etc.). You cannot use any of the other divisors within a millimeter because they are not evenly divisible with a single decimal.
Uncertainty is really the important factor in determining the number of significant figures. Significant figures for a number is determined by the number of digits physically shown on a measuring tool plus one estimated one. Using the picture below:
The gray bar is 11.66 cm ± 0.02 cm. Now the gray bar could also be 11.65 cm ± 0.05 cm. Some people might even say that it is 11.65 cm ± 0.01 cm, however, based on the size of the space between 11.60 cm and 11.70 cm, it seems unlikely that it can be subdivided 10 times equally using the naked eye. So for this gray bar, the number of significant figures in the measurement is 4. With the least significant figure being either a 6 or a 5 depending on the uncertainty of the measurement.
There are 4 basic rules for determining the number of significant figures. These rules are based on the type of number and its position within the actual measurement. The rules for the number of significant figures are:
 Nonzero numbers (numbers from 19) are always significant.
 Sandwiched zeros (zeros between nonzero numbers) are significant.
 Trailing zeros (zeros at the end of a number) are significant if there is a decimal point.
 Leading zeros (zeros at beginning of a number) are NOT significant.
Examples:
Number 
Signicant Figures 
Rule 
12345 
5 
1 
1020304 
7 
2 
12.40 
4 
3 
0.00565 
3 
4 
