I just don't get the t-test I know it sounds abysmally ignorant, but here we go.
I understand the basic logic behind the t-test: you want to know whether the difference in the mean of two samples is due to chance or not. To this end, you need to take variability into account. But variability at which level?
Suppose I want to run a dependent t-test on an experiment I've made. I have 20 athletes who each run 10 races without EPO, then 10 races with EPO in their blood. My question is whether EPO makes any difference in their performance.
I have two levels of variability in my data: intra-subject variability (the difference in performance between two races run by the same athlete) and inter-subject variability (the variability in the athlete's average performances).
What shall I do? Compute the average performance of each athlete for each condition, and then compute the t-test from this, or work directly from the raw data?
 A: I don't think this is so abysmal.  It takes some sophistication to recognize this discrepancy if you are primarily familiar with the $t$-test.  There are actually two levels of dependency in your situation: the same runners are assessed in both of two conditions, and there are multiple (viz, 10) measures of each runner in each condition.  Let's start with some simpler studies than the one you conducted and work our way up:  


*

*Randomly assign each runner to either the EPO group or the no EPO group and have them run the race once.
This is a two, or independent, samples $t$-test

*Have each runner run the race twice (hopefully with enough time in between to recover fully), but once with EPO and once without.
This is a paired, or dependent, samples $t$-test  

*Randomly assign each runner to either the EPO group or the no EPO group and have them run the race ten times.
This is an ANOVA with one between subjects factor (EPO vs. not) and one within subjects factor (race).  

*(Your design) Have each runner run the race 20 times, 10 with EPO and 10 without.
This is an ANOVA with two within subjects factors (EPO and race).  

