Data are called paired when you choose to compare the subjects of your study two by two instead of comparing one whole group to another whole group. The data from the two groups will be compared with a paired test.
Paired data are usually used in case-control studies but also in other study types as before-after studies or correlation analyses.
In the case of case-control studies, the main advantages of this type of design are to control chosen confusion biases and to reduce the within-subject variability. It may also need a smaller sample size than others study types. The inconveniences of this design are a) that the data used for matching cannot be studied if the matching were performed at the participant selection, b) a bad matching will lead to biased conclusions and c) the cutoff of numeric data to paired people could be difficult to find.
In the case of before-after studies and correlation analyses, the data are analyzed from the same patients. For example, you could look for a correlation between the age of patients and their systolic blood pressure. In this case, each age is matched with the same patient's systolic blood pressure.
To do a Student's paired test or a Wilcoxon's Rank Signed test
To do a McNemar's test