Still confused with the p-value definition 
The p-value is defined as the probability, under the assumption of the
  null hypothesis H, of obtaining a result equal to or more extreme
  than what was actually observed.
    A p-value lower than a
  threshold level of significance α means either that the null
  hypothesis is true and a highly improbable event has occurred, or that
  the null hypothesis is false.


Does this definition apply to any single measured data point of the sample or only to the "mean" (statistics) of the sample?

I've always thought this methodology is confusing.
This question is not exactly the same as other p-value questions you can find at stackexchange.
 A: "A result" doesn't mean much out of context—read "test statistic". You choose a test statistic to condense the  data into a single number that indexes its discrepancy with the null hypothesis in some direction of interest to you; you calibrate that test statistic by asking how often, from hypothetical repetition of a relevant sampling or random assignment procedure, you'd get values as extreme as, or more extreme than, the one you in fact got.
The value of a single observation $x$, or a function of it,† can be used as a test statistic. Sometimes the same test statistic as you'd use in a larger sample (for a sample size $n$, let $n=1$) will be useful, e.g. the mean or the maximum; sometimes not, e.g. the standard deviation or the range.
† Because more extreme values of $x$ don't always imply greater discrepancy with the null hypothesis.
A: *

*A test statistic is a function that takes all of the data as input and gives you a single number as the output (usually).

*If the null hypothesis is true, then someone can derive the statistical distribution of the (population) test statistic. 

*If you get a sample test statistic (the test statistic that's calculated using your data) that, according to this known probability distribution, is deemed not very likely, then you reject the null.

*The p-value is the probability of getting a sample test statistic that is as much or more extreme than the one that you got in real life; the probability is calculated using the known probability distribution of (population) test statistic conditional on the null hypothesis.

*This methodology makes no sense, yet since everyone uses it, you need to understand it anyway.

