I think the question needs to be split into two parts – what is a triple and what makes an “RDF triple” so special?
Firstly, a triple is, as most of the other commenters here have already pointed out, a statement in “subject/predicate/object” form – i.e. a statement linking one object (subject) to another object(object) or a literal, via a predicate. We are all familiar with triples: a triple is the smallest irreducible representation for binary relationship. In plain English: a spreadsheet is a collection of triples, for example, if a column in your spreadsheet has the heading “Paul” and a row has the heading “has Sister” and the value in the cell is “Lisa”. Here you have a triple: Paul (subject) has Sister(predicate) Lisa (literal/object).
What makes RDF triples special is that EVERY PART of the triple has a URI associated with it, so the everyday statement “Mike Smith knows John Doe” might be represented in RDF as:
uri://people#MikeSmith12 http://xmlns.com/foaf/0.1/knows uri://people#JohnDoe45
The analogy to the spreadsheet is that by giving every part of the URI a unique address, you give the cell in the spreadsheet its whole address space….so you could in principle stick every cell (if expressed in RDF triples) of the spreadsheet into a different document on a different server and reconstitute the spreadsheet through a single query.
Edit:
This section of the official documentation addresses the original question.