Transmon qubits are superconducting qubits built from a Josephson junction shunted by a large capacitor. The capacitor suppresses the circuit's sensitivity to charge noise, which was the main limitation of the earlier Cooper pair box design. Transmons are the workhorse of most large-scale superconducting quantum processors today, including those from IBM and Google.
The circuit behaves like a nonlinear LC oscillator. The Josephson junction supplies the nonlinear inductance, and the shunt capacitance sets the charging energy. The two lowest energy levels of this anharmonic oscillator are used as $\lvert 0 \rangle$ and $\lvert 1 \rangle$.
$$H = 4 E_C (n - n_g)^2 - E_J \cos\varphi$$
Here $E_C$ is the charging energy, $E_J$ is the Josephson energy, $n$ is the Cooper pair number operator, $n_g$ is an offset charge, and $\varphi$ is the superconducting phase across the junction. The transmon regime is defined by $E_J / E_C \gg 1$, which flattens the energy bands against $n_g$ and makes the qubit frequency nearly immune to charge fluctuations, at the cost of a smaller but still useful anharmonicity.
Qubit state is manipulated with microwave pulses at the qubit's transition frequency, typically in the 4-8 GHz range, delivered through a capacitively coupled drive line. Readout is done dispersively: the qubit is coupled to a microwave resonator whose frequency shifts depending on the qubit state, and that shift is measured by probing the resonator.
Coherence times for transmons are typically in the tens to hundreds of microseconds, limited mostly by dielectric loss in the materials surrounding the junction.