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
You are trying to apply significant figure rules to numbers with uncertainty, but you actually don't need to think about significant figures when the uncertainty is explicitly reported.
The purpose of significant figures is to loosely indicate the uncertainty in the measurement. In this case, like you are suggesting in your question, if we just said the length is $5.65\,\mathrm{cm}$ then this means we are sure about the $5.6$ part, but we aren't exactly sure about the $0.05$ part. In relation to the measuring device, with the given tick marks we are essentially saying, "My ruler has ticks every $0.1\,\mathrm{cm}$, so I'm sure this pencil is longer than $5.6\,\mathrm{cm}$. However, I don't have tick marks more precise than this, so I'm going to use my best guess based on my vision and ability to line things up to say it is half way between the $5.6$ and $5.7$ tick marks at $5.65\,\mathrm{cm}$." Based on this, it is obvious that the final significant digit tells others that we are not sure about that last $0.05\,\mathrm{cm}$. Even the math we do with keeping track of significant figures and the rules for addition vs. multiplication with using the correct amount of significant figures is to make sure we keep consistent with the uncertain digits.
The issue though is that significant figures don't tell us how uncertain we are. Do we think we have amazing sight and eatimation so we are within $0.01\,\mathrm{cm}$ of the $5.65\,\mathrm{cm}$? Or are we less confident in the $5.65\,\mathrm{cm}$ and want to report a larger uncertainty? Significant figures don't allow us to do this. Therefore, we can instead explicitly report the uncertainty as $5.65\pm0.05\,\mathrm{cm}$. At this point we don't need to worry about what that extra $0.05\,\mathrm{cm}$ in the measurement represents. The reported uncertainty takes care of this for us. And there are rules for doing math with numbers with uncertainty that keep our uncertainties consistent with the operations we choose to use, just like we have for significant figures.
Therefore, you don't need to worry about significant figures if uncertainty is explicitly reported. The uncertainty already tells us what is actually certain and what is not, thus it replaces (and improves) what significant figures are supposed to do.