First of all, it is crucial to always keep in mind that quantum mechanics cannot be used for superluminal communication, and neither to store more information than the classical counterparts allows.
This said, quantum mechanics does allow for different kinds of communications. For example, QKD protocols allow the sharing of a private key between two parties in a way that ensures that no eavesdropper is intercepting the channel (under a bunch of assumptions, as always).
It is also in principle possible to exploit the "entanglement link" that can be created between two parties to double the communication rate of the channel, via what is called superdense coding, essentially by transmitting two bits with a single qubit (but destroying the shared entanglement in the process).
A different aspect of essentially the same phenomenon is quantum teleportation, which is a protocol to transmit one qubit using two bits (and assuming to share an entangled pair beforehand).
While quantum teleportation has very little in common with the common conceptions of "teleportation", it is a very important aspect of quantum mechanics, and for example is an important aspect of many paradigms of quantum computation (KLM and one-way QC being the first that come to mind).
Regarding more specifically the quantum internet, I quote this recent nature news article on the subject.
The authors distinguish six steps that will likely happen in the process of building a full-blown "quantum internet":
SIX STEPS TO A QUANTUM INTERNET
Researchers have laid out six stages of sophistication that a future quantum internet could reach, and what users could do at each level.
0 Trusted-node network: Users can receive quantum-generated codes but cannot send or receive quantum states. Any two end users can share an encryption key (but the service provider will know it, too).
1 Prepare and measure: End users receive and measure quantum states (but the quantum phenomenon of entanglement is not necessarily involved). Two end users can share a private key only they know. Also, users can have their password verified without revealing it.
2 Entanglement distribution networks: Any two end users can obtain entangled states (but not to store them). These provide the strongest quantum encryption possible.
3 Quantum memory networks: Any two end users to obtain and store entangled qubits (the quantum unit of information), and can teleport quantum information to each other. The networks enable cloud quantum computing.
4 & 5 Quantum computing networks: The devices on the network are full-fledged quantum computers (able to do error correction on data transfers). These stages would enable various degrees of distributed quantum computing and quantum sensors, with applications to science experiments.
As you can see, the authors seem to state that the ultimate goal is to perform quantum computation on these networks.
A naive take on the idea is that the advantages given by quantum computation rely crucially on the coherence (think the entanglement) between the various qubits being very well maintained. Then, if one wants to perform some sort of "parallel quantum computation" in a meaningful way, that would require to transmit the qubits coherently, which requires to have a quantum network allowing this sort of thing.
It should be noted that this is all still quite speculative at this stage. To quote the last paragraph in the linked article:
Researchers’ opinions are not unanimous as to whether these applications will truly be useful, or whether a quantum internet will ever be sophisticated enough to make them broadly available. But some are optimistic. “I have no doubt that it will exist at some point,” Wehner says. But, she adds, “I think it is going to take a long time”