Last week it was reported in the popular press that a new study had found a rise in the number of younger women (ages 25-39) who were being diagnosed with advanced breast cancer.  I did not seek out the actual research, but what I did hear on the news (likely NBC Nightly News) is important to clarify.  We were told that in the last 30 years, the rate of diagnoses per 100,000 women went from 1.5 to just about 3.  That means for every 100,000 women in that age group, almost 3 were found to have advanced breast cancer in 2009.  The news reporters and scientists have noted that this is a small but significant increase in cases. (the significance statement is being questioned by some researchers, but lets say it is not).
   When you read a study and see that it is significant what that means is that it is 'real'.  It does not mean it is big or even important. You need both the effect size and whether or not it was 'real' to make your assessment.  Is 1.5 more cases a big deal?  That is for you to decide. In the statistics world, the significance means that the finding (here more cases of breast cancer) is very likely due to something other than chance. 
    When comparing two values (here at two time points) the assumption is that there is no difference.  The p value tells you the percent of times you could expect to find a real difference if there  wasn't one.  How many times you'd be making a mistake.
  The study researchers think that the finding in this case is real.  It could be that more women are obese and that obesity somehow increases the risk, or that as women are starting puberty earlier this could be upsetting hormone levels and increasing the risk.  The scientists who published this current study do not KNOW the reason.  They are only saying that the cases increased and something is behind it.  (Usually scientists or statisticians set a level of .05 or .01 at the start of a study and only accept 'significance' if they have a p < .05).   I am thinking you wish I had stopped talking about a paragraph ago!
Just eat right and exercise, ok!?  ( and of course, don't expose yourself to cigarette smoke, first hand or otherwise)