You use a matched-pair t-test if you're looking at 2 measurements within the same individual, group, condition, etc.
I.e, comparing before and after times of how long it takes a rat to run a maze after giving some kind of experimental treatment.
I'm not sure what you mean by "non-pooled" T test, but I think you might be getting at a Independent samples t-test. This is basically what it sounds like. You compare two different groups on some variable.
i.e, you have two groups of mice. One group is female, one group is male. You give some sort of experimental treatment then measure how long it takes them to run through the maze. You compare Males vs Females to see if they differ.