Significant figures are extremely important when performing any sort of calculation that involves a measured quantity.  This is because there is a limited degree of precision in any measurement that we take and the result of our mathematical manipulations of those figures cannot be more exact than the initial measurement.  If I use my bathroom scale to find the mass of a bowling ball, it might tell me that the bowling ball is 6kg.  That doesn't mean it's 6.0000 kg, that just means that it's somewhere between 5 and 7 kg.  This also means that when we perform calculations, we have to round off our final answer so that it does not convey more accurate information than is actually known.

Following are some rules that apply to the correct reporting of significant figures.

Counting Significant Figures:
In many cases, the number of significant figures is just the number of digits in the number.  For example:
2.83 has 3 sig. figs
5623 has 4 sig. figs
The hard part comes when we have zeros.  Are zeros significant?  The answer is: maybe.  If a zero appears between some non-zero numbers, it is significant.  So in a number like 23045, the zero is significant and the number has 5 sig. figs.  But not all zeros appear between non-zero numbers.  We call these trailing and leading zeros. 

Trailing and leading zeros are not significant figures, they are simply telling us where the decimal point is supposed to go.  For a number like 325000, the 3 trailing zeros do not convey any information, they are simply telling us where the decimal point should be and this number has only 3 sig. figs.  On the other side, the number 0.000325 also has only 3 sig. figs, the leading zeros just tell us where the decimal point should be.  There's got to be a trick here, right?  Well, not really.  The only place where we might get mixed up is when we have digits that look like trailing zeros that are actually significant.  What if we have a number like 100.00?  You might be tempted to say "hey, those are just a bunch of trailing zeros, so this number only has 1 sig fig".  Not quite.  Any zeros that come to the right of the decimal point in this type of a number are there to tell us that ALL the digits are significant.  This number has 5 significant figures.  In fact, just the presence of the decimal point is enough to tell us that what look like trailing zeros are actually significant figures.  The number "23000" has 2 significant figures, but the number "23000." has 5 sig figs.  Similarly for small numbers, "0.0023" has 2 sig figs (the leading zeros are not significant), "0.0023000" has 5 (the leading zeros are still not significant, but the zeros to the right of "23" wouldn't be there if they weren't sig figs).

How do we keep things straight?  The easiest way is by using scientific notation.  When a number is expressed in scientific notation, the decimal point is placed by the power of 10, so we NEVER have insignificant trailing or leading zeros.  If a number is properly expressed in scientific notation, ALL of the figures are significant.  A proper scientific notation includes a number from 1 to less than 10 and a power of 10. Expressing all of the numebrs from the preceding paragraph in proper scientific notation:
325000 = 3.25x105 (3 sig figs)
0.000325 = 3.25x10-4 (3 sig figs)
100.00 = 1.0000x102 (5 sig figs)
23000 = 2.3x104 (2 sig figs)
23000. = 2.3000x104 (5 sig figs)
0.0023 = 2.3x10-3 (3 sig figs)
0.0023000 = 2.3000x10-3 (5 sig figs)

Exceptions to Counting Significant Figures:
When we talk about sig figs, we are always referring to quantities that are measured because every measurement has a limited accuracy and precision.  If a number is defined rather than measured, we have an unlimited number of sig figs.  The normal boiling point of pure water is defined to be 100 degrees Celcius.  This is a definition.  The normal boiling point of water is 100.0000000000000000000000 degrees Celcius.  Similarly, the density of water is defined to be 1 g/mL.  That's a definition, so the density of water is 1.000000000000000000 g/mL.

There is also a special class of measurements for which the number of sig figs is unlimited.  If we are explicitly counting a group of items, the number that we count does not have a limited accuracy.  If I count 52 students in the class, that does not mean that the number of students is somewhere between 51 and 53, it means that there are 52.000000000000 students in the class.  This is why the number of significant figures is not limited to 1 when we consider the stoichiometry of reactants in a chemical equation; if 1 "Molecule A" reacts with 1 "Molecule B" in a balanced equation, those are exactly 1 and 1, not some number that's close to one.

When adding or subtracting 2 numbers, the answer cannot be more precise than the less precise number.  For example:
100.01 + 0.10001 = 100.11
790 + 0.153 = 790

Whenever you are multiplying or dividing 2 numbers, the answer can have no more significant figures than the number of significant figures in the number with fewer significant figures.  That's a mouthful.  In short, if you're multiplying or dividing a number that has 2 significant figures and a number that has 8 significant figures, your answer cannot contain more than 2 significant figures.  For example:
543.21 x 0.012 = 6.5

Calculations that contain both addition/subtraction and multiplication/division:
This is where trouble usually starts, but you just need to follow all the rules step by step.  The key here is that you NEVER round off intermediate numbers throughout your calculation, you only round off the final answer so that it has an appropriate number of sig figs.

What about pH?
This one gets a little tricky.  pH is a logarithm, so technically speaking, only digits after the decimal point are significant because the number before the decimal point only gives the power of 10.  This means that a pH of 6.82 has only 2 sig figs.  But it also means that if a pH of 9 is reported, that number seems to contains NO significant figures.  (It actually has 1 sig fig, but that's still not much.)  To understand this a little more clearly, it helps to know how logarithms work and understand that pH is just a logarithmic expression of the [H3O+] (or [H+]).

The important thing is to practice, practice, practice. If you always try to use proper significant figures in everything you do, it will become habit and you won't have to think about it. Good luck.