This one is relatively simple. Very similar names for two totally different concepts.
Hierarchical Models (aka Hierarchical Linear Models or HLM) are a type of linear regression models in which the observations fall into hierarchical, or completely nested levels.
Hierarchical Models are a type of Multilevel Models.
So what is a hierarchical data structure, which requires a hierarchical model?
The classic example is data from children nested within schools. The dependent variable could be something like math scores, and the predictors a whole host of things measured about the child and the school.
Child-level predictors could be things like GPA, grade, and gender. School-level predictors could be things like: total enrollment, private vs. public, mean SES.
Because multiple children are measured from the same school, their measurements are not independent. Hierarchical modeling takes that into account.
Hierarchical regression is a model-building technique in any regression model. It is the practice of building successive linear regression models, each adding more predictors.
For example, one common practice is to start by adding only demographic control variables to the model. In the next model, you can add predictors of interest, to see if they predict the DV above and beyond the effect of the controls.
You’re actually building separate but related models in each step. But SPSS has a nice function where it will compare the models, and actually test if successive models fit better than previous ones.
So hierarchical regression is really a series of regular old OLS regression models–nothing fancy, really.