A Noble Gas is a group of elements that in their standard state have a filled electron cloud.
These elements are found in the 18th column of the periodic table and include Helium (He), Neon (Ne), Argon (Ar), Krypton (Kr), Xenon (Xe) and Radon (Rn). They are all odourless and colourless mono-atomic elements.
Because these elements are already electron stable and do not need to gain or lose electrons in order to complete the valence electron clouds, these elements are relatively inert or un-reactive. However, adding an electric charge to these elemental gases causes the illumination used in tube (Neon) lighting systems.
Noble gas electron configurations are easier to write because they are shorter than full electron configurations.
To understand the chemistry of an element, it's most important to know about the electrons in the highest energy levels.
For example, the element magnesium has valence electrons (the two electrons in its highest energy level). This fact means that magnesium will form ionic compounds by donating two electrons. The magnesium ion has a +2 charge in ionic compounds (examples include MgO and #MgCl_2#)
Lead has four valence electrons and its chemistry is based on this fact. Many lead compounds result from lead atoms donating 4 electrons to form +4 ions (#PbO_2# and #PbCl_4# are examples).
The Noble gas shortcut electron configuration is a way of summarizing the information about the electrons of an atom which shows only the electrons most relevant for understanding the chemistry of the element.
Here is a video which discusses how to write the Noble gas shortcut electron configuration for magnesium and lead.