Snapping based on a single vertex

Is there any way to select a mesh inside edit mode (such as a circle) and select one of the vertices on the circle and snap that vertex onto a vertex of another mesh (such as the corner of a cube)?

Yeah… if you change snapping mode to Vertex, that should be the standard behavior.
The snapping options are on the right side of the 3d-view menu bar, or you can use ctrl+shift+tab to shange snapping mode. Then hold down ctrl to snap.