For example, we have 10 people whose BP is checked before medication. 2 hours after the medication, the BP of the same 10 patients are checked to see the effect of medication. This is a matched pairs t test, as the populations in question are the same. In a matched pairs test, the sample size has to be the same.
When the underlying populations are different, then we use a 2 sample ( t or z ) test. For example, a set of 25 students of AAA college had SAT scores of 560 and another set of 30 Students in BB college have SAT scores of 495. Now the populations here are completely different, and therefore we do not use a matched pairs t test