The elements of $H(\kappa)$ are the sets that are hereditarily of cardinality less than $\kappa$. If $x\in H(\kappa)$, then $|x|<\kappa$, $|y|<\kappa$ for every $y\in x$, $|z|<\kappa$ whenever there are $x$ and $y$, such that $z\in y\in x$, and so on.
This gives the intuitive idea, but it’s not really a definition. For that it’s easiest to start by defining the transitive closure of a set $x$:
$\operatorname{tr cl}(x)=\bigcup_{n\in\omega}{\bigcup}^n(x)\;,$
where
${\bigcup}^n(x)=\begin{cases} x,&\text{if }n=0\\\\ {\bigcup}\left({\bigcup}^{n-1}(x)\right),&\text{if }n>0\;. \end{cases}$
Then $H(\kappa)=\{x:|\operatorname{tr cl}(x)|<\kappa\}\;.$
(Although it doesn’t have the form required by the axiom schema of comprehension, this definition can be justified by showing that $H(\kappa)\subseteq V_\kappa$ for every infinite cardinal $\kappa$.)
Assuming AC, $H(\kappa)=V_\kappa$ iff $\kappa=\omega$ or $\kappa$ is strongly inaccessible.