An identity could be thought of as an equation where both sides are the same way or a different way of expressing the same thing. The interesting property about this is that it will hold true for no matter what values are chosen for the unknown(s). An identity could therefore be thought of as an equation with infinitely many solutions.
This will become 2a=2a when the left hand side is simplified. It is worth noting in simple English that this identity means when you add a number to itself like this, you will get two times that number.
A typical equation that is not an identity is an equation where the left hand side is not equal to the right hand side except when the correct value(s) of the unknown(s) is/are found, indeed if there is a solution.
5c -9=4c +7
The left hand side is not another way of expressing the right hand side and vice versa, therefore it is not an identity.
c=16 is the solution and will make both sides equal to 71.