Two-qubit system is a quantum system consisting of two qubits. The state space is the tensor product $\mathbb{C}^2 \otimes \mathbb{C}^2 \cong \mathbb{C}^4$, and a general two-qubit state is a superposition over four computational basis states: $\lvert 00\rangle$, $\lvert 01\rangle$, $\lvert 10\rangle$, and $\lvert 11\rangle$.
A general two-qubit state $\lvert\psi\rangle$ requires four complex probability amplitudes $c_{00}, c_{01}, c_{10}, c_{11} \in \mathbb{C}$ satisfying the normalization $|c_{00}|^2 + |c_{01}|^2 + |c_{10}|^2 + |c_{11}|^2 = 1$.
$$\lvert\psi\rangle = c_{00}\lvert 00\rangle + c_{01}\lvert 01\rangle + c_{10}\lvert 10\rangle + c_{11}\lvert 11\rangle$$
Two-qubit states that can be written as a tensor product $\lvert\psi_A\rangle \otimes \lvert\psi_B\rangle$ are called product states or separable states. If $\lvert\psi_A\rangle = a\lvert 0\rangle + b\lvert 1\rangle$ and $\lvert\psi_B\rangle = c\lvert 0\rangle + d\lvert 1\rangle$, their tensor product expands as:
$$\lvert\psi_A\rangle \otimes \lvert\psi_B\rangle = ac\lvert 00\rangle + ad\lvert 01\rangle + bc\lvert 10\rangle + bd\lvert 11\rangle$$
The notation $\lvert ab\rangle$ is shorthand for $\lvert a\rangle \otimes \lvert b\rangle$, where the left qubit is usually called qubit 0 (or the most significant qubit) and the right qubit is qubit 1.
Two-qubit states that cannot be written as a product state are called entangled states. An example is the Bell state $\lvert\Phi^+\rangle = (\lvert 00\rangle + \lvert 11\rangle)/\sqrt{2}$, which cannot be factored into any product $\lvert\psi_A\rangle \otimes \lvert\psi_B\rangle$. The four Bell states form an orthonormal basis for $\mathbb{C}^4$ and are the canonical maximally entangled two-qubit states. Entanglement is one of the key resources that enables quantum speedups.