Perhaps it will make more sense to explain how a D-Wave machine works, as (a) they actually exist and you can use one today for free online, and (b) it's way simpler than a gate model QC is.
Imagine a bunch of magnets. Imagine forcing them into a "frustrated" configuration; maybe you have them all on a grid, and you have servomotors that can rotate them to face any way you like. The servomotors are strong, so counteracting the magnetic forces is easy for them. You design an appropriately frustrated configuration, and then release all of the magnets at once. What configuration do they rotate into?
A quantum annealer is conceptually similar. Each qubit, on a regular, patterned graph, has connections to its neighbours. You can leave these alone (no corellation) or tune them up to +/- 1, corresponding to "must be the same as this other qubit" and "must be different than this other qubit". You can also bias each individual qubit to be a 0 or a 1.
Then, you let it go, and it anneals, and you observe the result. Your goal is to get to the _lowest energy state_ possible: the least possible frustration remaining.
In our magnet example, it would be as few magnets as possible wanting to move - if you poked them with your finger they'd want to go back into their current state. You could imagine that your magnets might not get down to their absolute lowest energy state; maybe it would take too much energy to flip from their starting state to that lower state. In a quantum system, because of tunneling, the system can reach these lower ground states. Rather than being in a fixed position the way our magnets were, qubits are in a quantum superposition, so they can reach a lower energy state without having to climb up that energy hill. Or so we're lead to believe by the numbers, anyway; I'm not a physicist.
Now, if you can map some useful computational question onto the original configuration of qubits that is answered by the ending position, you've got yourself a useful quantum computer. This is the hard part! The key is to use optimization algorithms where a lower energy state = a more optimized result. If you can do this, there's a ton of employment waiting for you.
Then, if you want "quantum supremacy", it's matter of providing more optimized answers in less time, particularly as the problem scales up in complexity. There does indeed appear to be a crossover point coming in a decade or so, at least for the small class of real-world problems that the Ising Hamiltonian works for.