What do we mean by positive terminal and negative terminal? What do we mean by positive terminal and negative terminal of  battery ? What is the electric potential of both positive terminal and negative terminal ? Does positive terminal mean positive electric potential energy and negative terminal mean negative electric potential energy  ?
 A: What do we mean by positive terminal and negative terminal of battery ? What is the electric potential of both positive terminal and negative terminal ? Does positive terminal mean positive electric potential energy and negative terminal mean negative electric potential energy ?
Yes that is correct when we consider "conventional" current, i.e., the flow of positive charge. 
These questions are actually interrelated and have to do with understanding why a "positive" terminal is considered to be at high potential and the "negative" terminal is considered to be at low potential. It has to do with how electrical current is defined. 
In electrical engineering current is considered the flow of positive charge. They call this "conventional current". This convention was established before current flow was fully understood. Physicists don't care for this, because for the most part (semiconductor current being an exception) current is the flow of negative charge (electrons). But that's the way it is and doesn't matter as long as one is consistent.
Now, for positive charge to flow from the negative terminal of a battery to the positive terminal work must be done because the force is repulsive. This increases the potential energy of the positive charge. Thus the positive terminal is considered to be at "higher" potential energy. This is somewhat analogous to raising a mass in the gravitational field. Work must be done to raise the mass against the force of gravity. The mass gains gravitational potential energy. 
If the positive charge moves from the positive terminal to the negative terminal then the force is attractive and the charge loses potential energy. So in that sense the negative terminal means negative potential energy.
But if we considered current to be the flow of negative charge (electrons), when electrons move from the negative terminal to the positive terminal, where the electrical force is attractive rather than repulsive, the electron loses electrical potential energy. So for electron flow, the positive terminal is lower potential energy. Conversely if they move from the positive terminal to the negative terminal, they gain potential energy. So the negative terminal would then be considered at higher potential energy.
Bottom line: Whether the positive terminal is considered to be higher or lower potential energy depends on whether current is considered to be the flow of positive charge or negative charge. In electrical engineering it is considered to be positive charge. But it could just as well have been the other way around.
Hope this helps.
