What does "nested" mean?

A nested design is one where the statistical analysis model treats the effects as an inseparable component of the other effects. In this case they're saying that the main effects of suicide and cardiovascular diseases near the time of diagnosis were confounded (inseparable) from the likelihood of committing suicide or dieing from cardiovascular disease at any time during the course of the disease. A simpler example of a nested design is where you are interested in whether boys or girls are more likely to show a particular trait or if it's more prevalent in one area of the country vs another. In this case gender and location would be your main effects but you'd also have to account for the fact that you can't separate gender and location into distinct groups so you'd add "gender nested within location" (written gender(location)) in the analysis model.

Excellent!
It's crystal clear.
Thank you.