Does anybody know how to explain briefly how to realize qubits (initialization, states transition, measurements,..) with superconducting Josephson junctions, say DC-SQUIDS?
I think the subject matter of supercondcuting qubits is rather broad and diverse, making it challenging to accurately capture it in a 'brief explanation'.
With that said, this recent review (Krantz et al., Applied Physics Reviews 6, 021318 (2019)) - "A Quantum Engineer's Guide to Superconducting Qubits" (arXiv:1904.06560) from the MIT group may be a good place to start.
Section II in the article is perhaps the most relevant to this question, as it describes several implementations of superconducting qubits, from the transmon and tunable transmon to flux qubits and fluxonium. The article further discusses, in sections IV and V, the state of the art in terms of control and readout.
The article does not discuss charge- and phase-qubits, which were abandoned (for the most part) because of their higher sensitivity to noise and material losses.