Moderate level definition: An operator is a mathematical symbol takes some defined number of inputs (also called arguments) and returns an output.
The first operator that students learn about is \(+\). This operator takes two values and gives their sum. For instance, \(5 + 4 = 9\). The arguments of addition are called the addends; the output is called the sum.
The second operator that students learn about is \(-\), which returns the difference of two values. For instance, \(5 – 4 = 1\). Addition is commutative, meaning that \(5 + 4 = 4 + 5\), but subtraction is not, so the arguments have different names. The first one is called the minuend (\(5\) in this example), while the second one is called the subtrahend.
Next comes multiplication, which takes two multiplicands and returns a product, and division, which takes a dividend and a divisor and returns a quotient.
At some point, students are also exposed to the unary operator forms of \(+\) and \(-\). Elementary textbooks will sometimes provide these in superscript form to disambiguate them from their binary versions, but in later grades the symbols are overloaded (to use computer programming parlance: a function is overloaded if it can take different numbers of arguments).
Multiplication has several conventional ways of being represented: \(\times\) is common early on, to be largely replaced by \(\cdot\). When extended character sets aren’t an option, * is often used. And sometimes multiplication is merely implied, with or without parentheses. So all of these are equivalent: \(5\times x=5\cdot x=5x=5(x)=5\text{*}x\).
Likewise, division has several conventional ways of being represented: \(\div\) is common early on, to be largely relaced with / when formal type-setting isn’t an option and the fraction bar when it is. So all of these are equivalent: \(8\div2 = 8/2 = \frac82\).
By the end of high school, most students have seen two other binary operators, in addition to a handful of functions. These two are exponentiation, which is traditionally indicated by the use of a superscript but can also be shown with a caret (^), so \(2^3 = 2\text{^}3\), and the radical, shown by \(\sqrt[3]{5}\) (for the cube root of five, as an example).
Additionally, the degree (\(^\circ\)) and percent (\(\%\)) signs are arguably unary operators, although a valid claim could also be made that they’re units.
There are other operators that many students see in high school, such as the unary operator !. Calculus introduces the operators \(\int\), \(\Sigma\), and \(\text{d}\), among others.
The difference between an operator and a function is largely a notational one: An operator is effectively a function that takes special formatting, generally for the purposes of making the notation faster to both write and read.
Other than the operators listed above, the most common functions in high school are the logarithm and three trigonometric functions. There is no particular reason why these could not be represented with operator symbols as well; it is mere historical accident that they are not.