# What does it mean “given a NFA and its accepted language L build the DFA that accepts L*”?

Supposing we have either a NFA or a DFA with accepted language $L$, what is $L^*$? And how can I build an automaton that accepts it?

$L^*$ is the Kleene closure of $L$. This is the language of all strings that can be made by taking a finite number of (not necessarily different) strings from $L$ and concatenating them. More formally,
$$L^* = \{\varepsilon\}\cup\{w\in\Sigma^*\mid \exists k\geq 1\,\exists w_1, \dots, w_k\in L \text{ such that } w = w_1\cdot\dots\cdot w_k\}\,.$$
Or, if you prefer, for $k\geq 1$, let $$L^k = \{w_1\cdot \dots \cdot w_k\mid w_1, \dots, w_k\in L\}$$ and then $L^* = \{\varepsilon\}\cup L^1 \cup L^2 \cup L^3 \cup\cdots\,$.
As for how to produce an automaton that accepts that, I suggest you think about it some more yourself, now that you know what it is you're trying to do. A hint would be to use nondeterminism to "guess" where the words $w_2, \dots, w_k$ begin.