Confidence Intervals and all that other confusing stuff!!

Thank goodness we are reviewing standard deviation, standard error, confidence intervals and the like. For some reason, it doesn't seem to matter how many times I have gone over this information in the past it always seems to disappear into some blackhole in my brain, it's so annoying. Confidence intervals, Type I and II errors and interpretation of the p-value are perfect examples of this, all of these seem to have really subtle differences between how you define what is right and what is wrong.

I found with confidence intervals the easiest way for me to understand it was that if 100 experiments were conducted, 95 of them would contain the population mean and 5 wouldn't, if alpha=0.05.
As for the p-value, definitions such as this : If the populations really have the same mean overall, what is the probability that random sampling would lead to a difference between sample means as large (or larger) than you observed? - just do not seem to make it crystal clear, why should we be concerned with the probability of events that have not occurred, so confusing! My interpretation: A small P value e.g. p<0.001 means that your observations are highly unlikely under the null hypothesis of no difference e.g. the mean number of umbrellas left on New Orleans buses is signficantly different from the mean number left on the Texas buses = small p value. If more umbrellas are left on buses in New Orleans then I guess we could infer that people from New Orleans are more forgetful ha ha!! In a nutshell, a small P value makes us reject the null hypothesis because an unlikely event has occurred were the null true. As for Type errors, how can I make this stick?? My interpretation : You said something was different when it wasn't - type 1, you said something was the same when it wasn't - type II. No doubt all this will be forgotten by this evening!!

P.S. Please correct my definitions if they are wrong :)