Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, vector optimization and order theory) show, there are many examples of convex cones with an empty (topological as well as algebraic) interior. In such situations, generalized interiority notions can be useful. In this article, we present new representations and properties of the relative algebraic interior (also known as intrinsic core) of relatively solid, convex cones in real linear spaces (which are not necessarily endowed with a topology) of both finite and infinite dimension. For proving our main results, we are using new separation theorems where a relatively solid, convex set (cone) is involved. For the intrinsic core of the dual cone of a relatively solid, convex cone, we also state new representations that involve the lineality space of the given convex cone. To emphasize the importance of the derived results, some applications in vector optimization are given.