Mp vs Mr (AISC Table 3-2)

Can anyone give me some insight as to what Mr and Mp are related to on table 3-2 of AISC?  I can't find information anywhere and it seems the practice problems I look at just magically pick which moment to use from this table.  Also, what is Z3 and how do I use it?

Thanks for the help!

Maybe someone can also explain the difference between yielding and ultimate.  When a question asks to find the strength of a problem, I really don't understand which equations to use.

check out Figure 3-1 

it all has to do with the unsupported length  (Lu) for the lateral torsional buckling limit state 
Mp is the plastic capacity (section will not experience any local buckling) it can achieve the plastic capacity; this occur when Lu <= Lp

Mr is the elastic capacity (section will experience some local buckling after the initial yielding,  Lp < Lu  < Lr

if Lu>Lr the section will experience local buckling prior to the yielding.

Hope this helps !