Expressions
An expression is collection of letters / numbers / mathematical operators without an equals sign.
An expression cannot be solved.
Equations
An equation is 2 expressions which equal each other, with the equal sign placed between the 2 expressions. This means that an equation can be solved for a particular variable.
Formulae (aka formulas)
A formula is a kind of equation which contains at least 2 variables(letters). The purpose of a formula is to show the relationship between the variables.
Identities
Identities are like equations but both sides of the equation always equal each other, no matter what values are chosen. You can always rearrange one side so that it is identical to the other side.
An identity should have a triple equals in the centre instead of a normal equals sign, but in reality this isn’t the case.