What is likelihood?

I am currently a TA for an introductory biostatistics sequence at JHSPH where we teach students about the essentials of regression analysis. A great question that came up at office hours last week was, “What is likelihood?” I love this question because it is so fundamental to statistical thought, seems very intuitive, but actually abounds in nuance. I found my answer to the question to be rather unsatisfying: “Likelihood refers to how probable our collected data would be given the regression model that we’re currently fitting.