Most quantum classes that are studied (like QMA, BQP, QIP, and exotic ones like QMIP and QRG) have a classical counterpart and you get the quantum class by thinking quantumly and changing your definition of "efficient computation" to "polynomial time on a quantum computer."
Often there are two steps to be made to reach from a well-known class to a quantum class. The first step adds randomness and error, and the next step adds quantumness. So for NP, the chain of classes is $NP \subseteq MA \subseteq QMA$. In NP the verifier is deterministic, and the certificate or proof is a bitstring. In MA, we modify the verifier to be a BPP machine (give it access to randomness and allow it 1/3 probability of error), while the certificate remains a bitstring. In QMA, we allow the verifier to be quantum and the certificate to be a quantum state. Since a quantum verifier can simulate a probabilistic one, QMA contains MA.
The easy way to get a quantum class out of most natural classical classes is to change one's notion of "efficient computation" from P or BPP to BQP, and change one's notion of information exchange from bits to qubits. So, for example, QIP is the same as IP when the BPP verifier is made BQP and the communication allows qubits instead of bits.
Finally, to get back to QMA: If you truly believed that BQP captures efficient computation in our universe, then QMA is the set of all problems that have an efficient verification procedure. That is, if you met an almighty prover, it would be able to convince you with high probability of $x \in L$ assuming you can handle qubits and do quantum computation. (And of course when $x \notin L$, there is almost nothing it could say to fool you.)