I saw from their Wikipedia page, and my impression is that both are developed when the data is nested. Are multilevel model and random effect model the same concept? If they are not, what are their differences and relations? Thanks!
Yes, to quote the Wikipedia page on multilevel model:
Multilevel models (also hierarchical linear models, nested models, mixed models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level.1
It naturally arises from observing the hierarchical and the clustered structure of the collected data typical found in treatment comparisons in clinical trials. It allows modelling residuals at different level, in other words, between-groups and within-group components for a two-level model.
I just re-iterated something that I learnt in my course, I could expand a bit on this if needed :)