Quantum X gate (or Pauli-X gate, quantum NOT gate) is a quantum gate that is analogous to the classical “NOT” or “bit flip”. For a single qubit, it applies a rotation around the X-axis by $\pi$ on the Bloch sphere. In matrix form, X gate is equivalently written as a Pauli matrix $\sigma_x$. This is why it's commonly called a Pauli-X gate.
$$X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$$
For states $\lvert0\rangle,\lvert 1\rangle$ on the computational axis (the Z-axis) it flips the state. This is why it's called a quantum analogue of “NOT” or “bit flip”. $$X\lvert 0\rangle = \lvert 1\rangle\qquad X\lvert 1\rangle = \lvert 0\rangle$$ The states $\lvert i\rangle, \lvert -i\rangle$ on the phase axis (the Y-axis) are flipped in the same way. $$X\lvert i\rangle = \lvert -i\rangle\qquad X\lvert -i\rangle = \lvert i\rangle$$ The states $\lvert +\rangle, \lvert - \rangle$ on the hadamard axis (the X-axis) are eigenstates of $X$ meaning those states are unaffected by the operator. $$X\lvert +\rangle = \lvert + \rangle \qquad X\lvert -\rangle = \lvert -\rangle$$