RO Core http://purl.obolibrary.org/obo/ro/core.owl is a subset of the full RO. It includes "core" relations intended to be domain neutral, and excludes most domain-specific relations (such as the "habitat of" relation used in environmental ontologies, or the "synapsed with" relation used in neuroanatomical ontologies).
Smith 2005 describes the origin of RO Core, with 10 very general relations used by ontologies across the OBO community. These relations are defined in terms of the Basic Formal Ontology (BFO), an upper-level ontology that includes very general terms such as 'entity', 'continuant', 'occurrent', 'material entity', 'process', 'disposition', etc. For instance, the 'has participant' relation holds between an occurrent (e.g. the process of growth of some frog) and a continuant (e.g. the frog itself). See ROAndBFO for more information.
RO Core imports a subset of classes from BFO 2.0, defines relations among them, and uses them for domain and range axioms. Every term in RO Core should have a textual definition and examples of usage. Since BFO is an upper-level ontology, not specific to biology, RO Core is also very general. The RO Core OWL file is available as a separate download from http://purl.obolibrary.org/obo/ro/core.owl. A number of OBO ontologies import RO Core specifically, when they do not need the wider range of biological relations provided by RO.
RO Core also diverges from BFO in that we do not include the BFO 'inheres in' relation. We instead have a relation characteristic of, which is in some aspects more general that inheres-in (it relaxes range constraints, such that things can be characteristics of processes), and in some aspects more specific than inheres-in (it restricts the relation to be functional).
Note that the RO characteristic of used to be called 'inheres in', as it was originally modeled after the RO relation, but was renamed in 2021 to avoid confusion. See this pull request for historic context.