What is the difference between an identity, an equation and a conditional equation? What is the difference between an identity, an equation and a conditional equation?

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Equation means equality. They are both related to the word equal. If such an equality is true for all values of the variable, it is called an identity, e.g., $\sin^2x+\cos^2x=1$ is true for all x. If however the equation in question only holds for some values, which one is supposed to determine, then it's called conditional, and its variable is termed an unknown.
Equation is a mathematical description which is equal only for one variable. But identity is a mathematical description which is always equal for any variable.