Significant figures in calculations I do understand the basic principle of significant figures which is to indicate how accurate the given result is.  
However I can't get it into my head why it should be a good idea to actually use them (Bare with me for a moment).  
If considering the following mesaurements:
$$
m = 0.25~\text{kg} \hspace{2cm} v = 0.25~\frac{\text{m}}{\text{s}}
$$
Each of the measurements has 2 significant figures.  
If I know want to calculate the momentum of the object (assuming above measurements are from the same object) I'd do
$$
p = 0.25~\text{kg} \cdot 0.25~\frac{\text{m}}{\text{s}} = 0.0625~\frac{\text{kgm}}{\text{s}}
$$
As can be seen the calculated result now has 3 significant figures. If respecting the rules of significant figures one would have to to say $p=0.063~\frac{\text{kgm}}{\text{s}}$ (which by the way would forbid the use of the equals sign which is always ignored aas far as I have seen).  
So in terms of showing to others that the third decimal place is uncertain one actually makes a not so precice result (due to the measured data) even worse by rounding it.  
Also if one would need to perform further calculations that include the calculated momentum wouldn't the very same kind of rounding error propagate through the whole calculation ultimately ruining the end-result completely (assuming that the calculation has many intermediate results that are all rounded to the respective significant figure)?  
To sum it up: In my opinion it would only make sense to use all digits one can get for calculate the result. After that one would need to perform a error calculation and then after having calculated the error of the result one can round the result because the result is given with the calculated error (e.g. $p = 0.063 \pm 0.001~\frac{\text{kgm}}{\text{s}}$).  
But simply using these "magically" significant figures for reasoning that certain digits have to be discarded doesn't seem logical to me.  
Furthermore doesn't a calculated result lose all its meaning when given without its error? So why do we need significant figures so that others don't get confused (taking the result as more accurate than it is)?  
Or are significant figures some sort of lazy rule of thumb if one doesn't want to calculate the error of one's result?
 A: First of all: always rounding '5' up will lead to a systematic bias (use a random number generator to convince yourself that this is so). To mitigate this, round '5' to the nearest appropriate even digit.
Assuming your 0.25 was actually some number on (0.245, 0.255), uniformly distributed, then the statistical uncertainty is $0.01 / \sqrt{12}$. (also verifiable with a uniform RNG).
Then, using the chain rule on $p=mv$ and adding the terms in quadrature (independent errors):
$$(\delta p)^2=(m\delta v)^2 + (v\delta m)^2$$
Ignoring units (for now, out of strict laziness):
$$(\delta p)^2=2(\frac{1}{4}\frac{1}{100 \sqrt{12}})^2=\frac{1}{6} \frac{1}{400^2} \approx 10^{-6}$$
so
$$\delta p \approx 0.001 \ \mathrm{kg\cdot m/s}$$
and you can safely claim:
$$p = 0.0625(10)\ \mathrm{kg \cdot m/s}$$
where the $n$-digits in parenthesis represent the uncertainty in the last $n$ digits.
A: I agree with nearly everything you said - and double emphasize that writing $p=0.063$ kg m/s is a violation of the equal sign. When I finish a calculation, I  always use $\simeq$ or something similar, and I believe that's the best practice to follow. However, significant figures give us a set of rules to use when dealing with numbers that allow us to set numbers equal to each other in ways which cannot set variables equal to each other.
But I would like to point one thing that you kinda danced around - errors are not the same as significant figures, and we should not expect them to do the same job. If you measure something, you should always give the error on the measurement, and not trust the significant figures to do that for you. When you do a calculation using measured values, you should always propagate the errors through the calculation (or use some other method if you have a population, for instance).
Significant figures do not tell you how accurate a particular number is. When you say "$m=0.25$ kg", you are telling me that's an absolutely perfect number. If I multiplied this by $0.999999999$ m/s (as a perfect number), is the answer really $0.2499999998$ kg m/s? Well maybe, but given that I started with 0.25 kg, the context is such that 0.25 kg m/s is clearly the answer (even though I should probably write $\simeq 0.25$ kg m/s).
Significant figures tell you which digits are significant in a particular situation, which allows you to use the equal sign in a way which we all understand. They are guidelines (so that we don't have to have this conversation every time we write a number), not estimates of error or uncertainty.
