A longitudinal study is somewhat similar to a repeated measures study but in this case, people are studying and restudied over a period of time (as opposed to studied across different experimental conditions as is the case with a repeated measures study). The longitudinal study design is good for looking at the effects or changes over a long period of time, usually as people age.
For example, a psychologist may study the effects of counselling for children of divorce as they age. The children may get counselling for one year and then the researcher measures the children on different things (e.g., coping, stress, anxiety, etc.) once a year for the next 10 years and compare them to children who did not receive the counselling, but were measured in the same way and times. This would allow the researcher to see if the counselling had any effects as the children aged compared to those who did not receive counselling.