How does further dividing the population increase confidence? I am reading up on significance tests but I can't quite grasp where the "number of groups" from my example fits in.
When I ask 100 people which party they would vote for, imagine I get these results:
Party A | Party B
-----------------
     53 | 47

This could mean a tendency in favor of party A or just be coincidence. Now I could divide the 100 people into 2 groups, do the poll and get these results:
Party A | Party B
-----------------
     26 | 24
     27 | 23
-----------------
     53 | 47

I'd say this gives more confidence that there is a tendency in favor of Party A. Even further divided:
Party A | Party B
-----------------
     18 | 15
     20 | 13
     15 | 19
-----------------
     53 | 47

On the other hand, the tendency would be doubtful if the results were like these:
Party A | Party B
-----------------
     18 | 15
     21 | 12
     14 | 20
-----------------
     53 | 47

How is this "number of groups" called and how can I see (mathematically) that the poll is useless if that number is 1.
 A: Well, you're subdividing your samples, presumably on some other variable (like say, race or age group or geographical location). As your sub-samples become smaller, the variance of the difference in proportions increases. Note that if the subgroups are not homogenous with respect to proportion, the assumptions under which the CI is usually derived won't hold.
Somewhat relevant: http://en.wikipedia.org/wiki/Margin_of_error
Such calculations are affected by the size of the subsample, not the overall sample.
When looking at a difference in proportions that add to 100%, the margin of error for the difference in proportions is substantially larger than the corresponding standard error for the difference of independent proportions. You can estimate covariances of proportions from the same sample using multinomial results.
Sometimes such subgroups are deliberately sampled individually: 
http://en.wikipedia.org/wiki/Stratified_sampling
Another useful point to consider when subdividing:
http://en.wikipedia.org/wiki/Simpson%27s_paradox (like almost everything in statistics, not discovered by the person it's usually named after)
Hope that's of some help in clarifying whatever it is you want.
