Basically, it means that the correlations could be used to send a message. Or simply that Bob’s measurement outcomes can reveal some details of Alice’s actions.
This is impossible when Alice and Bob each hold one qubit of a Bell pair. Despite the entanglement present, as well as contextuality, signaling in this case would result faster than light communication. And that’s not allowed.
But if Alice does something to a qubit and then gives it to Bob, signaling is certainly allowed. Alice could simply encode bit values as $|0\rangle$ and $|1\rangle$, for example.
The process of handing over the qubit is referred to as the ‘quantum channel’. This could be done by physically sending it, using a technique such as teleportation to transfer the state, or just Alice shouting ‘Hey Bob, this qubit is yours now’ across the lab. Often channels come with the application of noise, because nothing in life in perfect. But sometimes they are just a concept we invoke in our stories about Alice and Bob.
Note that non-signaling correlations usually mean that some source correlated a couple of qubits, and then sent one to Alice and the other to Bob. There is then no causal link between what Alice does to hers, and what measurement results Bob gets from his. This is the reason for the lack of signaling.
Signaling correlations, however, usually mean that Alice was the source of the qubit that was sent to Bob. So there is a causal relation: Alice directly affects what state Bob gets, and can use this to send information.
This is how we would usually want to describe things but this paper would not want to use such language. This is because it is probing situations where the causal order can be indefinite. It therefore tries to identify these different kinds of correlations in a more general way, and also see if there is anything else.