Getting into the full details of a triple store is far beyond my level of expertise. Think of this as the very not-technically savvy tl;dr of this book, which is worth a read if you really want to dig in on web semantics.
Something like our old friend
consists of three parts: a subject (Quotation), a predicate (publishedIn), and an object (Source).
A standard database (MySQL or its sexier descendants) purports to store relationships—they are called relational databases. But in practice, so-called relational databases store only the subject and the object, along with a key that allows for referencing across different tables. (For you Excel fans, it's like a pivot table on steroids.) But a relational database does not store the actual relationship—the predicate that explains how the subject and object are connected.
A triple store (sometimes also called a graph database or an RDF database), stores subject, object and predicate as triples (hence the name). That added bit of complexity makes triple stores far more powerful than a typical database. Indeed, because it stores the relationships between data, a well-designed triple store can actually surface new information by drawing inferences that aren't themselves explicitly stored as separate pieces of data.
For example, imagine a triple store database that contains the following data:
bio:John b:hasGrandchild b:Matt
bio:Joe b:hasChild b:Matt
bio:John b:hasChild b:Josh
bio:Joe b:hasGender b:cishet
Such a system could deduce that Josh is Matt's uncle, even though it does not directly store any information linking Josh and Matt. (The first three lines entail one of three relationships between Joe and Josh: partners, brothers or brothers-in-law. The fourth line rules out partners. Either of the other two relationships leaves Josh as Matt's uncle.)
This sort of thing is (at least part of) what it means when people talk about "machine learning." Indeed, Facebook's Social Graph, Twitter's Interest Graph and Google's Knowledge Graph are all proprietary examples of triple store databases. It's their ability to draw new inferences from stored data that makes them so valuable to advertisers (and so creepy to privacy advocates).
The digital gardening movement picked up steam with the release of Roam Research, a note-taking tool that incorporates automatic bidirectional links, using an authoring system that is more like a wiki than like a traditional note-taking app. Roam's linked notes surface new information—not unlike the graph database.
These kinds of conceptual organization schemes are known as ontologies, and they map terms in multiple dimensions. More importantly, they also formally document the relationships between terms. The predicate that links subject term with object term.