What is the meaning of $e^{int}$ notation

The exponent $int$ is indeed a product of the imaginery number $i$, an integer $n$ and a variable $t$. By Euler's formula, we have $$e^{int} = \cos(nt) + i\sin(nt).$$

What is important about this sequence of functions $\{\phi_n(t)\}_{n\in\mathbb{Z}}$ is that they form a orthonormal Schauder basis for the inner product space $L^2(0,2\pi)$, i.e.

$$\int_0^{2\pi} \phi_m(t) \overline{\phi_n(t)} \, dt = \begin{cases} 1 & \text{ if } m = n,\\ 0 & \text{ otherwise.}\end{cases}$$

and for any $f \in L^2(0,2\pi)$, there exists a sequence $(f_j)_{j=1}^\infty \subset \langle \phi_n \rangle$ such that $$\int_0^{2\pi} \Big( f(t) - f_j(t) \Big)^2 \, dt \to 0 \text{ as } j \to \infty. $$

I believe from context $\varphi_n(t)=\frac{1}{\sqrt{2\pi}}e^{int}=\frac{1}{\sqrt{2\pi}}(\cos(nt)+i\sin(nt))$

$i^2=-1$

$n \in \mathbb Z$

with argument $t$.