What is ATP and how does it function? What would happen if we did not have ATP?

ATP is short for adenosine triphosphate. It is the universal unit of energy used by biological cells, and it is produced in the mitochondria.  When a cell needs energy, it gets it from ATP.  The name itself tells a great deal about the structure of ATP.  A single ATP is composed of three parts.  The first part is a base.  In this case, the base is adenine.  The second part of the molecule is a sugar.  The sugar is a ribose.  The third part of the ATP molecule is a phosphate chain.  As the name implies, ATP contains three phosphates.  
The phosphate bonds are important because it is the phosphate bonds that carry a lot of potential energy.  When a cell needs energy, a phosphate is removed from the chain. When the bond breaks, the potential energy is released.  Some of my students have a hard time understanding how something "breaking" can release energy.  I illustrate it by stretching a rubber band until it breaks.  The potential energy of the stretched rubber band releases a lot of kinetic and sound energy when the rubber band snaps.  It's a decent analogy for how breaking the ATP molecule into an ADP molecule releases energy. 
I'm not sure which direction to go with the second part of the question.  "What would happen if we did not have ATP."  The short, simple answer is we would die.  Without ATP, cells wouldn't have their "energy currency" and would die.  All living things are made of cells, and as their cells die, the organism dies.  
On the flip side, if ATP never did exist, but life did manage to begin and exist anyway, I would have to say that life must be using some other form of molecular energy.  Then we would be studying that amazing molecule instead of ATP.  
http://www.chm.bris.ac.uk/motm/atp/atp1.htm