Ok, here we are speaking of non-interactive zero-knowledge proof systems for some language $L\in NP$. We there have a pair $\sf (P,V)$ of
probabilistic polynomial time algorithms (called the prover and the verifier) where both have input $x\in L$ and $\sf P$ additionally
holds a secret witness $w$ for membership of $x$ in $L$ and wants to convince a verifier $\sf V$ of this fact without revealing $w$.
One requires (a bit informal):
- correctness: for $x\in L$ all proofs generated by $\sf P$ will always be accepted by $\sf V$
- soundness: for $x\notin L$ proofs generated by by $\sf P$ will only be accepted with negligible probability by $\sf V$
- zero-knowledge: for $x\in L$ the view of a verifier can be efficiently simulated without
the knowledge of $w$. Actually, the definitions you provide are in the common reference string (CRS) model, i.e., there is a $crs$ to which all entities
have access to. For zero-knowledge one requires that $(crs,x,\pi)$ with $\pi \leftarrow {\sf P}(crs,x,w) $ and $(crs,x,\pi)$ with $(crs,\pi) \leftarrow {\cal S}(x)$,
where $\cal S$ is the so called simulator, are computationally indistinguishable.
Now, adaptive zero-knowledge means, that soundness and zero-knowledge are with respect to adaptive adversaries. For soundness this
means that an adversary $\cal A$ is allowed to choose $x\notin L$ after seeing the $crs$. For zero-knowledge one requires from
the simulator $\cal S$, that $\cal S$ has to output $crs$ first and afterwards produce the simulated proof $\pi$. These two steps of the
simulator are denoted as ${\cal S}^{crs}$ and ${\cal S}^{proof}$ respectively. Note that these two definitons
are stronger definitions than in the static setting (described in the beginning).
Since the authors are talking about zero-knowledge proofs of knowledge for some $NP$ relation $\cal R$, they additionally require a property known as knowledge extraction.
Basically this means that the witness $w$ can be extracted from the proof transcript (this extractor is the algorithm $E$).
They also use a common reference string $crs$ with a trapdoor $trap$ which is required in order for ${\cal S}^{proof}$ to simulate a
proof without knowing the secret witness (you may look for instance at Groth-Sahai NIZK proof system for examples).
The notation ${\cal A}^{S'(crs,trap,\cdot,\cdot)}$ means that the adversary $\cal A$ has oracle access to the algorithm $S'$ where the inputs
$crs$ and $trap$ are fixed and the latter two input arguments (indicated by $\cdot$) can be arbitrarily chosen by $\cal A$. Hope that helps a bit.
