A regression model predicts a value $Y$ given a set of $X$ variables. In the regression model $Y \sim X$, we measure the effect of $X$ on $Y$, i.e., we measure the association between changes in values of $X$ and changes in values of $Y$. Analyzing a regression model is a method used to solve a regression problem.

Reference: James, G., Witten, D., Hastie, T., & Tibshirani, R. (2017). An introduction to statistical learning: with applications in R. New York: Springer.