Problem with macbook battery

Something goes wrong with my battery.
My macbook pro is 2 years old.

Last days, the battery goes lower and lower.
When I put the power cord, does not charge at all, but just maintain the percentage of energy that already has!
When I pull out the power cord, the battery goes lower.
Then I connect the power cord again, once again, macbook pro does not charge, but it maintains the energy it has!

So it seems that my battery stop charging. It can only be stable at the current level or go down.

Can you please help me to solve this problem?

 

I dont use mbp running on its battery.
Everywhere I go, I attach it to the wall power socket.

Given the problem that I described above, If I disconnect mbp to completely run out of energy, in order to 'test' the battery, will it ever boot up or not?
Suppose that it will go to 0% battery. Then I will plug the power cord. It is possible that will not charge the battery, but it will have power supply. What do you expect to happen? Will I be able to use my mbp or not?

Can mbp run, even if the battery is in 0%, but attached to the power cord? Or it is needed to have a minimum of charge?