An identity is an equation that is true for all values of the variables. For example: The above equation is true for all possible values of x and y, so it is called an identity.
Strictly speaking we should use the "three bar" sign to show it is an identity as shown below. But it is very common to use the equal sign.
An identity is true for any value of the variable, but an equation is not. For example the equation is true only when x=4, so it is an equation, but not an identity. In fact, when we see an equation like that, we are usually trying to solve it. That is, find the single value of x that makes the equation true.
They are used in simplifying or rearranging algebra expressions. By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time.
For example, suppose we are working an algebra problem and we have the expression: We recognize this as right hand side of a familiar identity* So we can replace it with the thing on the other side of the identity:
In summary, an identity says that two things are equivalent. If you see one, you can replace it with the other.
Identities are only useful if you know them, since only then will you recognize that a replacement is possible. But there are a lot of them (see trig identities below). Get a feel for the common ones and have a quick reference handy to look them up.
There are dozens of identities in the field of trigonometry. For example, a popular one is: which is one of the so-called double angle identities.
For a full list seeTrig identities.