Just like you can understand the adjoint operator as the transpose of a matrix, or the normal operator as $z \overline{z} = \overline{z} z$ for $z \in \mathbb{C}$, how can I understand the compact operator on an easy example?
I can't seem to find any easy and intuitive examples on that. I've heard something about the integral (operators) being compact (why?), but that's all I know.
You want an easy example? Ok, there you go. The operator $T : \ell^2(\mathbb N)\to\ell^2(\mathbb N)$, defined by $$ (Tx)_n = \frac 1n\cdot x_n,\quad x\in\ell^2(\mathbb N), $$ is a compact operator. You can represent it as an infinite diagonal matrix with the sequence $(1/n)_{n\in\mathbb N}$ on its diagonal.
Compact operators are an important class of bounded operators. They are important because
So, the knowledge on compact operators often helps in the understanding of structures. The set of compact operators that are not finite-rank is just on the edge between the finite-dimensional and the infinite-dimensional world, where topology becomes more important.