I will try to address the question in the context of differential geometry of manifolds. A property of a manifold is termed _intrinsic_ if it depends only on distances as measured within the manifold. In a more mathematical language, intrinsic properties depends on the **metric** , or the **first fundamental form**. On the other hand, a property is termed _extrinsic_ if it depends on the way you embed your manifold in an higher dimensional Euclidean space. An example of extrinsic 'property' is the **second fundamental from** from which you can deduce the **shape operator**.
Another important example that may use the term intrinsic is the **curvature**. Gauss's **Theorema Egregium** asserts that the curvature is an intrinsic property of a manifold - because it can be deduced from the metric.