A function is a transformation or mapping of one thing into another thing. It might be written as a rule (e.g. "Take the input and square it"), as a formula ("e.g. $f(x) = x^2$ or $x \mapsto x^2$), as a set of ordered pairs (e.g. $\left\{(1, 1), (2, 4), (3, 9), \ldots\right\}$, or any other way of showing how the output relates to the input. The function doesn't have to use numbers, either - a function could take two words and return their letters interlaced (so f(cat, dog) = cdaotg) or it could tell you what day of the week a given date falls on, or the post code/zip code of a given geographical location.

[In very formal terms, a function is a set of input-output pairs that follows a few particular rules.]

An equation is a declaration that two things are equal to each other. For example, $2^2 = 4$ is an equation stating that the square of 2 is 4. An equation may include variables of unknown value, and it may be true for all, some or none of the possible values of those variables. For example, $x^2 = 4$ is an equation that is true when $x = \pm 2$, and false for other values of $x$, while $x^2 = -4$ is an equation that is false for all real values of $x$.

What may be confusing you is that we often use equations to declare a relationship between two variables, often in the form of a function or formula. For example, $y = x^2$ is an equation stating that the value of $y$ is determined by the value of $x$ via the function $x^2$.