Disclaimer: Quantum computing is really complicated, as are quantum physics in general. It's not something I know a lot about, and most of what I do know is from a single lecture. The following information is, as far as I know, correct, but heavily simplified.
Quantum computers operate on qubits, which are sort of like point in 2D space representing probability. The X axis indicates the probability of the value being 0, and the Y axis indicates the probability of the value being 1. For example, the coordinates (1.0, 0.0) would mean "definitely 0", (0.1, 0.7) would mean "probably 1", and (0.4, 0.4) would be indeterminate. Qubits remain in this probability-based state until something observes them, at which point they experience "wavefunction collapse" and appear in a single location.
Those probabilities can be adjusted by various logic gates, like traditional electrical values. The gates themselves and how they operate are somewhat different. Examples of them are CNOT, Pauli-X, and Hadamard gates. I'm not going to go into detail about how they work here, because I'm not confident enough in my understanding to be sure I would actually explain them correctly.
At this point, quantum computers are purely experimental, and haven't been scaled up to enough qubits to solve practical problems. Companies like Google, Microsoft, and D-Wave are actively researching them, but it's unknown when, if ever, they'll be able to start being used to real computing. Due to their intense operational requirements (total elelectrical and mechanical isolation and cooling to near absolute zero), it's unlikely that they'll ever see consumer use. Despite those challenges, research is continuing, because the impact of quantum computing on scientific and Google-scale computing could be immense.