T gate (or $\pi/8$ gate, or $\sqrt{S}$ gate) is a single-qubit gate that adds a phase of $\pi/4$ to the $\lvert 1\rangle$ state while leaving $\lvert 0\rangle$ unchanged. It is a special case of the phase gate with $\phi = \pi/4$, and is the square root of the S gate: $T^2 = S$.
$$T = \begin{pmatrix}1 & 0\\ 0 & e^{i\pi/4}\end{pmatrix}$$
On the computational basis, $T\lvert 0\rangle = \lvert 0\rangle$ and $T\lvert 1\rangle = e^{i\pi/4}\lvert 1\rangle$. On the Bloch sphere, $T$ is a $\pi/4$ rotation about the $z$-axis. Unlike the S gate, $T$ is not a Clifford gate: it does not map Pauli operators to Pauli operators under conjugation, making it the “hard” gate in fault-tolerant quantum computing. Implementing $T$ on error-corrected hardware requires magic state distillation, which consumes many physical qubits per logical $T$ gate. Together with the Hadamard gate and the CX gate, $T$ forms a universal gate set: any quantum computation can be approximated to arbitrary precision using only $\{H, T, \text{CX}\}$.