Is there a shortcut for switching between these three modes?

Is there a shortcut for switching between these three modes?

This can be accomplished by pressing Ctrl+Tab. This will bring up the shown popup, in which you can select either Vertex, Edge, or Face mode. Note that you can use V, E, and F or 1, 2, and 3 to complete the action using shortcuts.

It's 2019 ;)
Blender 2.80 became much more intuitive. The hotkeys for the select mode are now pre-configured.
Number keys 1, 2, 3 switch between vertex, edge, face selection mode.
Another important thing: If you have configured to emulate a numpad, you have to change the shortcuts from 1 to Numpad-1.
Search for mesh.select_mode in Preferences->Keymap
If you prefer to have own shortcuts, you can create custom ones in the Input section of Blenders User Preferences (File > User-Preferences > Input). Open the 3D View section, and add your shortcut to the Mesh section:
You'll need to fill in the path of the operator to call. For the Mesh selection, this will be view3d.select_component. Once done, define the keystroke you like, and choose a type from the dropdown box which appeared after setting the operator. Note: Even if the dropdown box already says 'Vertex', choose the option again. The box will be greyed out until you do that, and the keystroke won't function. Save User Settings after that, and enjoy your own keymap!
It's 2018 so let me add the shortcut command name for Blender 2.79:
mesh.select_mode is what you need for toggling vertex/edge/face in edit mode.
You want to add it under User Preference -> Input -> 3D View -> Mesh as described by this answer
I was asking the same questions when tried to learn Blender editor. I know that such a feature exists in the Lightwave 3D editor, but for some reason develepers of Blender forgot about it. Hopefully, Blender allows to add missing features with add-ons.
For the third version of Blender 3D editor you can use this add-on:
https://github.com/neverwinter-nights/Mesh-Selection-Mode-Cycle
It adds two actions:
Key combinations can be assigned to these actions to your liking.