I encounter significant digits much more often in chemistry than in physics. So basically:
Say you have a ruler with centimeter and millimeter markings. You measure the length of a pencil, and it comes out to somewhere in between 8.6 cm and 8.7 cm. It seems a touch closer to 8.6 than to 8.7. So, you say that the pencil is 8.63 cm long. The last digit implies that it is $\pm.01$. This way, the value could be 8.62, 8.63, 8.64, or anywhere in between. The most that you know is that it is definitely closer to 8.6 than 8.7, and the range from 8.62-8.64 just about covers your uncertainty about the measurement.
If you wanted to be absolutely precise, every single measurement you make and quantity you calculate would have a tolerance based on the limitations of your measuring apparatus. Of course, it would be cumbersome to keep writing $\pm.01$ every time, so it is simply assumed that the value is known exactly except for the last digit, which is uncertain.
Now when you do calculations, you can't use the value that you found, because it has some uncertainty associated with it. To be correct, you would have to carry out multiple calculations, first on the lower bound, and then on the upper bound, to figure out what the uncertainties of your new quantity are. This doubles the number of calculations you need to make, and is just cumbersome and tedious. That is why the rules for significant digits arose. They are a guideline for figuring out what kind of uncertainty your new quantity has without having to make any extraneous calculations. Thus, when you multiply 2 numbers, one with 3 sig figs and the other with 2, you will know for sure that the product will have 2 sig figs, one of which is absolutely certain, the other slightly uncertain.
To reiterate, in the example above, our value of 8.63 cm for the length of the pencil has 3 significant digits; two of which are absolutely certain (8 and 6), and the last one certain to $\pm1$.
NIST has its own more sophisticated guidelines for reporting uncertainty of measurments.
http://physics.nist.gov/Pubs/guidelines/TN1297/tn1297s.pdf
There is nothing wrong with reporting two digits in an uncertainity, and many peer reviewed journal articles do. This is especially true when the first digit of the uncertainity is $1$. If you report $\pm 1$, $1$ could mean anything between $0.5$ and $1.5$, which is not reasonably specific for the purpose of a published physics measurement. On the other hand $\pm 9$ is much more specific. Another way to think of this is, what if you express the uncertainty in base $2$ instead of base $10$? Then, $\pm 1$ would be $1$ digit and $\pm 9$ would be $4$ digits.
Best Answer
This depends on the context. If you have 1500 of something and you counted them yourself and you are sure you have precisely 1500, then all four figures are significant. On the contrary, if you're guessing that you have 1500, implying a certainty of order 100, then only the leading two figures are significant. In scientific notation, one would write the former as $1.500\times 10^3$ and the latter as $1.5\times 10^3$. Explicitly stating the trailing zeros indicates that those figures are indeed significant.