To expand on that, you can also use what I believe are called the object buttons to move objects to a new layer… the layer buttons will look exactly the same as what you posted, but be inside the object buttons, and they will move the selected object to a new layer in the same way as the m button. You MUST do this in object mode, however, in edit mode it will invoke the mirror command.
Layers are used in Blender the same as in photoshop or gimp, so that you can store objects you want out of the way in a different place, temporarily invisible.
it is on the header of the 3d window (as apparenlty you have seen)
objects can be on one or more layers, the one on the 3d window header selects which layers are visible
it is also in the environment map texture (to select materials to NOT put in the environment map), and in the object [formerly in edit] buttons (to select the layers an object is in)
layers are numbered, the top 10 are 1 through 10, the bottom 10 are 11 through 20 (left to right)
the seperation between the two sets is merely to make the choices more visible, it is difficult to count to 10, so layer 6 could otherwise easily be confused with 5
you can select layers by left clicking on them (to choose just one) or shift+left clicking to toggle. AT LEAST ONE LAYER MUST ALWAYS BE SELECTED
you can also use the numbers on the top of your keyboard (unless you set the option in the info buttons to treat those keys as the number pad, this may have laptop mentioned near the setting or on the tooltip)
each number key selects the layer 1 through 10 (0 is for 10), alt+the number selects 11-20. Shift+the number (or shift+alt+num) can be used to toggle layers
Well, actually, layers in Blender are not like layers in Photoshop or GIMP. In those programs, layers are like transparent plastic sheets placed in front of one another parallel to the image surface. Whereas in Blender (outside of the Sequence Editor), layers are simply a way of selectively hiding things as well as a way to control the effect of a layer-constrained lamp.