In logic, the order is roughly the number of collections over which you can use quantification.
For example, the first-order logic suppose that you have a lone "container" of elements, called Universe, over which you can quantified universally (for all or $\forall$) or existentially (there exists or $\exists$). The set theory ZF use only one order of quantification : all elements in the Universe are set. You cannot speak about another "type" of element. This is why you must use schemas to provide a "set" (in the usual meaning) of axioms. The axiom schema of restricted comprehension is one of them :
$$ \forall y \exists z \forall x (x\in z \Leftrightarrow (x\in y \wedge \phi(x)) $$
for all well-formed formulae $\phi$.
Here, you can see that we do not write
$$ \forall\phi\forall y \exists z \forall x (x\in z \Leftrightarrow (x\in y \wedge\phi(x)) $$
You can wonder why we use the literate formulation "for all well-formed formulae $\phi$" instead. This is because, quantifiers $\forall$ and $\exists$ only apply to set. There is no quantifier that can be apply on formulae.
If we really want them, we have to provide some new quantifiers that can be applied on another "collection" of elements; namely the well-formed formulae. These quantifier will be named "second-order quantifiers" and the language needed to express the new formulae will be referred as a second-order language. It is needed to be able to distinguish both order of quantification and the usage is to write the new operators with $\forall^2$ and $\exists^2$. Considering this, the second-order formula over sets and well-written first-order formulae would have been written as
$$ \forall^2\phi\forall y \exists z \forall x (x\in z \Leftrightarrow (x\in y \wedge\phi(x)) $$
There is some problems that can spawned from using quantifier over formulae (circular definitions for examepl) this is why it is not achieve this way, and why the literate formulation is preferred.
However, there exist other examples of second-order language. To keep logic examples, we can talk about von Neumann–Bernays–Gödel set theory that uses quantifications over set and proper classes, or Computation Tree Logic (CTL) that uses quantification over paths and states.
Hope this is helping you.