Ontology --- A model of the entities in a certain domain which makes explicit the types of entities and the relationships between them. This term came up in the context of a question about whether our grammar can connect to ontological semantics. The answer is that, while we don't do so directly in this course, the predicate names in the semantics good in principle be linked to an ontology.
Generative Grammar --- In the original sense, a generative grammar was one that explicitly accounted for a a language as a set of strings, but licensing strings in the vocabulary of the language which are acceptable and failing to license strings from the same vocabulary that are not. Sometimes it is used in a narrower sense to denote mainstream Chomskyan grammar. There is also a use of "generative" in machine learning which distinguishes "generative models" from "discriminative models". The former produce structures or analyses while assigning probabilities to them, while the latter only select among structures or analyses provided by some other means.
Metonymy --- Referring to an entity by using an expression denoting a related entity. The example in question was The roof admired the woman which is pragmatically fine in a context where the roof can be understood to refer, for example, to a group of people standing on the roof. See Wikipedia on metonymy.
Feature Structures --- Bundles of information expressed as pairs of features and values. Values can be atomic (symbols like '+' or 'thick' or types like adj) or feature structures themselves.
HPSG --- Head-driven Phrase Structure Grammar. A framework or general theory of formal syntax. The formalism and general theory developed in the course textbook are derived from HPSG, but we can't say they are HPSG, because a) there isn't one completely shared version of HPSG across the research community and b) the framework in the textbook is simplified for pedagogical purposes.
Saturation --- This is part of the idea that lexical heads enter the syntax with a set of requirements of what else should appear. Moving up the tree, as they find these other phrases, those requirements are 'canceled off' or 'saturated'. Inspired by the same metaphor from chemistry as 'valence'.
Head/Headedness --- The idea that constituents have many of their properties determined by some distinguished word inside of them (the "head").
Specifier --- A notion which groups together determiners of noun phrases and subjects of sentences.