Loosely speaking, ontologies are categories of objects defined by their attributes and relationships to other things. Where a hierarchy is a branching structure where items can only appear on the tree once, ontologies do not require everything to stem from a single "root" node and items can appear in the tree in more than one place.
It's a way of working around how hierarchies can't model some things very well. E.g. "bipedal" is an attribute that can apply to both animals and robots; where does it go in a hierarchy that it can apply to both animals and robots without also implying that robots are animals or vice versa.
Domain Driven Design - one of those things like Agile that triggers all sorts of holy wars - has a lot of overlap with the general concept of ontologies, to the point that I've seen some teams formalize all communication between microservices through a shared "ontology", which in reality was essentially a giant XML based descript of valid nouns and verbs that events could use to communicate between services.
Additionally, there's a good deal of overlap with the "semantic web" concept, which itself had a good deal of hype with very limited (but important) application- even the W3C has some published content on how all three fit together: https://www.w3.org/2001/sw/BestPractices/SE/ODA/
Maybe more in philosophy and classic general AI. Basically ontologies are systems of categorizing and classifying knowledge. E.g. if you want to reason about self driving, you would have an ontology that lets you separate traffic signs from billboards.
What does that have to do with this?
Is there some use of "ontology" in logic I have not heard of?