DrLecter's comment should really be an answer, but one point which bears emphasis is that it is not sufficient for crypto purposes to say "a (cyclic) group of prime order $p$", because even though mathematically all groups of order $p$ are essentially the same group (i.e., they are isomorphic), we can't really say the same thing in crypto.
When doing crypto in cyclic groups, we generally make the assumption that the discrete logarithm problem is difficult to solve in the groups in question (most often implicitly through a stronger assumption like those in the DH family). But the choice of a particular cyclic group makes a dramatic difference in the difficulty of solving the discrete logarithm problem, at least in the current state of our knowledge: we know cyclic groups of prime order in which the discrete logarithm problem is easy (which ones?), and others in which it seems to be difficult. And even among the groups in which the discrete logarithm problem is considered difficult, it seems to be more difficult in some than in others.
Hence, any discrete log-based cryptosystem needs to specify explicitly which group it operates in if we want to have any idea of its security. This is what $G$ and $G_t$ represent in the definition of a pairing: they are simply the groups in question (e.g., $G$ is the group of points of the elliptic curve with equation [...] generated by the point [...]).
EDIT: Of course, in abstract discussions we can simply say that $G$ is a cyclic group of prime order in which such-and-such assumption holds, but still we would need to be more specific than just saying we operate in cyclic groups of prime order. It is also very helpful in discussions and proofs to have a name for the groups we are working in, especially in pairing settings where there are two (sometimes three) groups involved.