First, decide whether the hypothesis you're considering is directional or non-directional. Here's how I usually think about it: As an experimenter, you're usually predicting that there will be some sort of difference between two conditions. If the difference the experimenter expects to find is non-directional -- for example, if I think that students who take a test while listening to jazz music will score differently than another group that isn't listening to any music, but I have no idea whether they'll do better or worse -- then the null hypothesis is simply the hypothesis that I'm wrong and there's no significant difference between the two groups after all. (A significant difference between two groups is a difference that's unlikely to be due to chance.) In other words, the null hypothesis would predict that the experimental group will have a mean score that is NOT significantly higher or lower than that of the control group.
If the difference the experimenter expects to find is directional -- for example, if I predict that kids listening to classical music will score BETTER than a control group that listens to no music -- then the null hypothesis is simply the hypothesis that I'm wrong and the kids listening to classical won't score significantly better (e.g., that they'll do the same, or possibly worse). Here, the null hypothesis predicts that the experimental group will NOT have a significantly higher mean score than the control group.
Next, take a close look at the way the results are stated. If you see phrases like any of the following:
"The experimental group had a significantly greater mean score than the control group...
We were able to reject the null hypothesis...
We were able to reject H0 ..."
...then these are all different ways of saying that there WAS a significant difference between the groups in question.
Conversely, if you see any of the following, it tells you that there was NOT a significant difference:
"The experimental group did not have a significantly greater mean score than the control group...
We failed to reject the null hypothesis...
We failed to reject H0 ..."
This does not necessarily mean that the research hypothesis is false, but merely that the possibility that there was no difference between the average scores of the groups cannot be conclusively rejected.
Why the convoluted language about "failing to reject the null hypothesis," rather than just saying "accept the null hypothesis"? The reason is that failing to reject the null hypothesis does not mean that the null hypothesis is true. In other words, returning to the example from Step 1, even if my classical music experiment didn't find a significant difference between the two groups, this does not mean I can claim that classical music has no beneficial effect on test scores. It might still have had an effect that my experiment was not sensitive enough to find. For example, my sample size might have been too small--if I had used more subjects, I might have been able to acquire a significant result.