Semiparametric probability models are a general class of models used for estimation and inference that contain a nonparametric component and a parametric component.
David Cox coined the term semiparametric when formulating the seminal Cox model and describing the role of the baseline hazard. At that time, several generalizations of the likelihood function existed, such as quasi likelihood, conditional likelihood, and partial likelihood, with little theory to reconcile their unique approach to inference and handling nuisance parameters. Semiparametric models did just that.
This allowed all such models, such as generalized linear mixed models, cox models, and conditional logistic regression to be written as the product of a parametric component and a non-parametric component. The latter of which can be "partialled" or "conditioned" out to obtain parametric values that can be compared on a likelihood-like scale.
Estimation in semiparametric inference has several properties that are similar to those of maximum likelihood such as root-n consistency, unbiasedness, and existence and uniqueness of solutions.
