Both problems require you to find a minimal subset of the vertices satisfying some covering condition. However, the difference is that the vertex cover problem requires you to cover all the edges, whereas the the dominating set problem requires you to cover all the vertices. To clarify what "cover" means, an edge is covered iff at least one of the edge's 2 vertices is in the set, and a vertex is covered iff it is in the subset or if it neighbours a vertex which is in the subset. For an example, see the attached image (stolen from the slides). All of the vertices are covered since each of the vertices have a neighbour in the subset (the vertices coloured black). On the other hand, the edges aren't all covered since the "bottom" edges do not have any of their vertices in the subset.