In the 3D View header, to the right of where you choose whether you are in the Global or the Local mode, there are three boxes, as seen in the following screenshot. The left one has an orange dot next to a cube. It is the Vertex Select Mode button. The center one has an orange vertical line next to a cube. It is the Edge Select Mode button. The right one has an orange parallelogram on a cube. It is the Face Select Mode button. They control whether you are selecting vertices, edges, or faces.