The Euler characteristic can refer to a number of different numerical invariants of different sorts of objects. But all of them bear a family resemblance and in some cases they can be shown to agree.
The Euler characteristic of a topological space can be defined as the alternating sum of the ranks of its homology groups (when this is finite).
The Euler characteristic of a dualizable object in a symmetric monoidal category is the trace of its identity morphism. Applied to suspension spectra in the stable homotopy category, this recovers the usual Euler characteristic of a manifold.
There is a definition, due to Tom Leinster, of the Euler characteristic of a (suitably finite) category, which generalizes the notion of groupoid cardinality.
Mike: Can the Euler characteristic of a category be recovered as the trace for a dualizable object in some symmetric monoidal category?