I am trying to make my plane exactely the width / height of a grid cell. I thought I could do that by clicking "Snap Element"->"Increment", but that does not do anything. Can anybody tell me what I am doing wrong? Thank you.
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You can snap elements to the nearest grid intersection with ⇧ ShiftS Snap to Grid:
Note that while in an aligned view (Top, Left, etc.) the subdivision level of the grid to which the mesh snaps will depend on the zoom. In an other view the mesh will snap to the displayed grid size.
The Snap to Increment mode of transform snapping will snap to grid increments. In other words, if your object/vertex is offset from the grid initially then it will stay offset.
By default the grid elements are 1 BU by 1 BU (Blender Unit), and the default plane is 2 BU by 2 BU. So to scale and position it to the grid, you can: